Concatenation-Type Selfie Numbers With Factorial and Square-Root

Inder J. Taneja

RGMIA Research Report Collection, 20(2017), pp. 1-42, http://rgmia.org/v20.php

Concatenation-Type Selfie Numbers With Factorial and Square-Root Received 13/09/17

Inder J. Taneja1

Abstract Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, squareroot, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have written selfie numbers by use of concatenation, along with factorial and square-root. The concatenation idea is used in a very simple way. The work is limited up to 5 digits. Work on higher digits shall be dealt elsewhere.

Based on Matt Parker’s [10, 11] idea of Concatenation to solve "The 10,958 Problem" [17].

Contents 1

Crazy Representations 1.1 Selfie Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Concatenation-Type Selfie Numbers 2.1 Sequential Representations . . . 2.1.1 Both Ways . . . . . . . . . 2.1.2 Digit’s Order . . . . . . . 2.1.3 Reverse Order of Digits . 2.2 Non Sequential Representations . 2.2.1 Both Ways . . . . . . . . . 2.2.2 Digit’s Order . . . . . . . 2.2.3 Reverse Order of Digits .

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2 3 3 4 4 4 5 7 7 10 17

3

Number Patterns

31

4

Summary: Selfie Numbers 4.1 Factorial . . . . . . . . . . . . 4.2 Factorial and Square-Root . . 4.3 Fibonacci Sequence . . . . . . 4.4 Triangular Numbers . . . . . 4.5 Binomial Coefficients . . . . . 4.6 S-gonal numbers . . . . . . . 4.7 Centered Polygonal Numbers

34 34 34 35 37 37 38 39

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Professor of Mathematics, Federal University of Santa Catarina, 88.040-900 Florianópolis, SC, Brazil. Web-site: http://www.inderjtaneja.com; E-mail: [email protected].

RGMIA Res. Rep. Coll. 20 (2017), Art. 138, 142 pp

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Inder J. Taneja

1


Crazy Representations

In [17], author wrote number from 1 to 11111 in increasing and decreasing orders of 1 to 9 and 9 to 1. By use of basic opeartion, we got all numbers except 10958 in the increasing case. We can write it by use of factorial and/or square-root. See below: 10958 := 1 + 2 + 3!! + (−4 + 5! + 6 − 7) × 89 10958 := 1 × 2 × (3!! − 4! × (5 + 6) + 7! − 8 − 9) For more solutions refer [5, 6, 7] Moreover, this work appeared as improbable research [1, 2, 3, 4]. Recently, Matt Parker [10, 11, 12] gave a new idea of finding this number by using concatenation among numbers. See below: 10958 := 1 × 23 + ((4 × 5 × 6)k7 + 8) × 9, where akb := 10 × a + b, a ∈ Z, b ∈ {0, 1, 2, 3, 4, 6, 7, 8, 9} .

(1)

The notation ” k ” is known as concatenation. More on it can be seen in [9]. The number 23 can also be written as 2k3, but it don’t make any effect on the representation. From mathematical point of view the study of concatenation as operator is very deep. For details see [9]. Some general idea can be see at [13, 15]. Some times it is called as Triangles of the Gods [16] given below: 1 12 123 1234 12345 123456 1234567 12345678 ... ... Remark 1.1. Our aim here is not to go much deep. Only to work with basic idea, i.e., 10 × a + b, where a and b are integers, with condition that b is only of single digit. Let us see below some examples, 2k3 := 2 × 10 + 3 = 23 (2 + 3)k5 := (2 + 3) × 10 + 5 = 55 (5 × 4 + 7)k(2 + 3) := 27 × 10 + 5 = 275. Here we should observe that, in case of real concatenation, when the second component of the pair is of two digits, then the first one is automatically of two digits, i.e., it become as 5k(7 + 8) = 5k15 = 5 × 100 + 15 = 515. Another example, 1k23 := 100 + 23 = 123. In this case we can say that 5k15 = 51k5 1k23 = 12k3. This type os situation is still not under study. In another opportunity, we shall work with general case of concatenation. Based on above details we shall bring selfie numbers. First, below is an idea of selfie numbers with examples. 2

Inder J. Taneja

1.1


Selfie Numbers

Numbers represented by their own digits by certain operations are considered as "selfie numbers". Some times they are called as wild narcissistic numbers. There are many ways of representing "selfie numbers". They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, sgonal values, centered polygonal numbers, etc. Below are some examples with factorial and square-root written in both ways, i.e., in digit’s order and its reverse: √ √ = 6! + (3!) 9 936 := ( 9)!3 + 6! q √ = 6( 9+2−1) 1296 := (1 + 2)!9 /6 √ √ 2896 := 2 × (8 + ( 9)!! + 6!) = (6! + ( 9)!! + 8) × 2 √ √ 331779 := 3 + (31 − 7) 7+9 = 9 + (7 × 7 − 1)3 × 3 √ √ 342995 := (34 − 2 − 9) 9 − 5 = −5 + (−9 + 92 − 4)3 √

759375 := (−7 + 59 − 37)5 = (5 + 7 + 3 ) 9−5+7 √ 759381 := 7 + (5 × 9)−3+8 − 1 = −1 + (8 × 3 − 9)5 + 7 More examples of selfie numbers in different situations are given as summary in last section 4. In this work our aim is to bring selfie numbers by use of concatenation, along with factorial and square-root. The work is limited up to 5 digits. Study for higher digits shall be dealt elsewhere.

2

Concatenation-Type Selfie Numbers

This section brings the selfie numbers by use of formula (1) with factorial and square-root. It has been divided in small subsections according to each type of situation. Before starting there are few numbers those can be written without use of factorial and square-root. See below:

• Digit’s Order 15129 := h(−1 + 5) k 1i2 × 9 33124 := h(3 × 3) k 1i2 × 4

• Reverse Order of Digits D E 1255 := 5 × 52 k 1

15477 := 77 × h(4 × 5) k 1i 24964 := (4 + h(6 + 9) k 4i)2

1288 := 8 × h(8 × 2) k 1i 1359 := 9 × h(5 × 3) k 1i 1449 := 9 × h(4 × 4) k 1i

26896 := (h(6 + 9) k 8i + 6)2 39304 := (h4 k 03i − 9)3

3


Inder J. Taneja

2.1

Sequential Representations

There are many numbers those can be written in sequential way, i.e., from 0 to 9 they have symmetric consecutive representations. Again we have divided it in three part. First one both ways, second one in digit’s order and third one in reverse order of digits. 2.1.1

Both Ways D 30960 := (3 + 0!) D 30961 := (3 + 0!) D 30962 := (3 + 0!) D 30963 := (3 + 0!) D 30964 := (3 + 0!) D 30965 := (3 + 0!) D 30966 := (3 + 0!) D 30967 := (3 + 0!) D 30968 := (3 + 0!) D 30969 := (3 + 0!)

2.1.2

Digit’s Order D √ 37440 := 3!! × 7 − 4 D √ 37441 := 3!! × 7 − 4 D √ 37442 := 3!! × 7 − 4 D √ 37443 := 3!! × 7 − 4 D √ 37444 := 3!! × 7 − 4 D √ 37445 := 3!! × 7 − 4 D √ 37446 := 3!! × 7 − 4 D √ 37447 := 3!! × 7 − 4 D √ 37448 := 3!! × 7 − 4 D √ 37449 := 3!! × 7 − 4

D√ E √ E 9 × 6! + 0 = 0 + 6! × 9 + 0! k 3 D√ E √ E k 9 × 6! + 1 = 1 + 6! × 9 + 0! k 3 D√ E √ E 9 + 0! k 3 k 9 × 6! + 2 = 2 + 6! × D√ E √ E k 9 × 6! + 3 = 3 + 6! × 9 + 0! k 3 D√ E √ E 9 + 0! k 3 k 9 × 6! + 4 = 4 + 6! × D√ E √ E 9 + 0! k 3 k 9 × 6! + 5 = 5 + 6! × D√ E √ E k 9 × 6! + 6 = 6 + 6! × 9 + 0! k 3 D√ E √ E 9 + 0! k 3 k 9 × 6! + 7 = 7 + 6! × D√ E √ E k 9 × 6! + 8 = 8 + 6! × 9 + 0! k 3 D√ E √ E 9 + 0! k 3 k 9 × 6! + 9 = 9 + 6! ×

k

√ E k 4 +0 √ E k 4 +1 √ E k 4 +2 √ E k 4 +3 √ E k 4 +4 √ E k 4 +5 √ E k 4 +6 √ E k 4 +7 √ E k 4 +8 √ E k 4 +9

39840 39841 39842 39843 39844 39845 39846 39847 39848 39849

4

:= := := := := := := := := :=

−(−3 + 9)! + 8! + h4! k 0i −(−3 + 9)! + 8! + h4! k 1i −(−3 + 9)! + 8! + h4! k 2i −(−3 + 9)! + 8! + h4! k 3i −(−3 + 9)! + 8! + h4! k 4i −(−3 + 9)! + 8! + h4! k 5i −(−3 + 9)! + 8! + h4! k 6i −(−3 + 9)! + 8! + h4! k 7i −(−3 + 9)! + 8! + h4! k 8i −(−3 + 9)! + 8! + h4! k 9i


Inder J. Taneja

46695 := 4 + 66 +

− h4! k 0!i + 0! + 8! + 0 − h4! k 0!i + 0! + 8! + 1 − h4! k 0!i + 0! + 8! + 2 − h4! k 0!i + 0! + 8! + 3 − h4! k 0!i + 0! + 8! + 4 − h4! k 0!i + 0! + 8! + 5 − h4! k 0!i + 0! + 8! + 6 − h4! k 0!i + 0! + 8! + 7 − h4! k 0!i + 0! + 8! + 8 − h4! k 0!i + 0! + 8! + 9 D √ E 44640 := (4!/4)! × 6 k 4 + 0 D √ E 44641 := (4!/4)! × 6 k 4 + 1 D √ E 44642 := (4!/4)! × 6 k 4 + 2 D √ E 44643 := (4!/4)! × 6 k 4 + 3 D √ E 44644 := (4!/4)! × 6 k 4 + 4 D √ E 44645 := (4!/4)! × 6 k 4 + 5 D √ E 44646 := (4!/4)! × 6 k 4 + 6 D √ E 44647 := (4!/4)! × 6 k 4 + 7 D √ E 44648 := (4!/4)! × 6 k 4 + 8 D √ E 44649 := (4!/4)! × 6 k 4 + 9

40080 40081 40082 40083 40084 40085 40086 40087 40088 40089

:= := := := := := := := := :=

46690 := 4 + 66 + 46691 := 4 + 66 + 6

46692 := 4 + 6 + 46693 := 4 + 66 + 46694 := 4 + 66 + 2.1.3

D√ D√ D√ D√ D√

9k0

E

9k1

E

9k2

E

9k3

E

9k4

E

6

46696 := 4 + 6 + 46697 := 4 + 66 + 6

46698 := 4 + 6 + 46699 := 4 + 66 +

69120 69121 69122 69123 69124 69125 69126 69127 69128 69129

:= 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k := 6! × h9 k

75840 75841 75842 75843 75844 75845 75846 75847 75848 75849

D√ D√ D√ D√ D√

9k5

E

9k6

E

9k7

E

9k8

E

9k9

E

(1 + 2) ! i + 0 (1 + 2) ! i + 1 (1 + 2) ! i + 2 (1 + 2) ! i + 3 (1 + 2) ! i + 4 (1 + 2) ! i + 5 (1 + 2) ! i + 6 (1 + 2) ! i + 7 (1 + 2) ! i + 8 (1 + 2) ! i + 9

:= 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4! := 7! × 5!/8 + h4!

k 0i k 1i k 2i k 3i k 4i k 5i k 6i k 7i k 8i k 9i

Reverse Order of Digits 01320 01321 01322 01323 01324

:= 0 + (2 + 3)! × h1 k 0!i := 1 + (2 + 3)! × h1 k 0!i := 2 + (2 + 3)! × h1 k 0!i := 3 + (2 + 3)! × h1 k 0!i := 4 + (2 + 3)! × h1 k 0!i

01325 01326 01327 01328 01329

5

:= 5 + (2 + 3)! × h1 k 0!i := 6 + (2 + 3)! × h1 k 0!i := 7 + (2 + 3)! × h1 k 0!i := 8 + (2 + 3)! × h1 k 0!i := 9 + (2 + 3)! × h1 k 0!i


Inder J. Taneja

03050 03051 03052 03053 03054 03055 03056 03057 03058 03059 03720 03721 03722 03723 03724 03725 03726 03727 03728 03729

:= 0 + 50 × h3! := 1 + 50 × h3! := 2 + 50 × h3! := 3 + 50 × h3! := 4 + 50 × h3! := 5 + 50 × h3! := 6 + 50 × h3! := 7 + 50 × h3! := 8 + 50 × h3! := 9 + 50 × h3!

D √ 21965 := 5 + 6! × ( 9)! D √ 21966 := 6 + 6! × ( 9)! D √ 21967 := 7 + 6! × ( 9)! D √ 21968 := 8 + 6! × ( 9)! D √ 21969 := 9 + 6! × ( 9)!

k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i

D 45360 := 0 + 6! × 3! D 45361 := 1 + 6! × 3! D 45362 := 2 + 6! × 3! D 45363 := 3 + 6! × 3! D 45364 := 4 + 6! × 3! D 45365 := 5 + 6! × 3! D 45366 := 6 + 6! × 3! D 45367 := 7 + 6! × 3! D 45368 := 8 + 6! × 3! D 45369 := 9 + 6! × 3!

:= 0 + (−2 + 7)! × h3 k 0!i := 1 + (−2 + 7)! × h3 k 0!i := 2 + (−2 + 7)! × h3 k 0!i := 3 + (−2 + 7)! × h3 k 0!i := 4 + (−2 + 7)! × h3 k 0!i := 5 + (−2 + 7)! × h3 k 0!i := 6 + (−2 + 7)! × h3 k 0!i := 7 + (−2 + 7)! × h3 k 0!i := 8 + (−2 + 7)! × h3 k 0!i := 9 + (−2 + 7)! × h3 k 0!i

04790 04791 04792 04793 04794 04795 04796 04797 04798 04799

:= 0 − 9 + 7! − h4! := 1 − 9 + 7! − h4! := 2 − 9 + 7! − h4! := 3 − 9 + 7! − h4! := 4 − 9 + 7! − h4! := 5 − 9 + 7! − h4! := 6 − 9 + 7! − h4! := 7 − 9 + 7! − h4! := 8 − 9 + 7! − h4! := 9 − 9 + 7! − h4!

D √ 21960 := 0 + 6! × ( 9)! D √ 21961 := 1 + 6! × ( 9)! D √ 21962 := 2 + 6! × ( 9)! D √ 21963 := 3 + 6! × ( 9)! D √ 21964 := 4 + 6! × ( 9)!

k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i k 0!i

k k k k k k k k k k

E k 1 /2 E k 1 /2 E k 1 /2 E k 1 /2 E k 1 /2

√ √ √ √ √ √ √ √ √ √

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

5+4

E

E √ 4 k8 D E √ 84961 := 1 + 6! × 9 + 4 k 8 D E √ 84962 := 2 + 6! × 9 + 4 k 8 D E √ 84963 := 3 + 6! × 9 + 4 k 8 D E √ 84964 := 4 + 6! × 9 + 4 k 8 D E √ 84965 := 5 + 6! × 9 + 4 k 8 D E √ 84966 := 6 + 6! × 9 + 4 k 8 D E √ 84967 := 7 + 6! × 9 + 4 k 8 D E √ 84968 := 8 + 6! × 9 + 4 k 8 D E √ 84969 := 9 + 6! × 9 + 4 k 8 84960 := 0 + 6! ×

E k 1 /2 E k 1 /2 E k 1 /2 E k 1 /2 E k 1 /2

6

D

9+


Inder J. Taneja

2.2

Non Sequential Representations

In this subsection, there are numbers those are not in a sequential way as of subsection above. Again, we have divided it in three subsections. One on both ways. Second on digit’s order and third on reverse order of digits. 2.2.1

Both Ways

Below are selfie numbers those can be written in both ways, i.e., in digit’s order and its reverse. The first three numbers are written in one way as they are palindromes. 15851 := (1 + 5!) × h(8 + 5) k 1i D √ E 44644 := (4!/4)! × 6 k 4 + 4 74347 := 7 × 43 × h4! k 7i √ E D √ E 396 := 3! k 9 ! ×6 = 6 k 9 ! × 3! D

10582 := h1 k 0!i × (5! × 8 + 2) = (2 + 8 × 5!) × h0! k 1i 10584 := h1 k 0!i × 5! × 8 + 4! = 4! + 8 × 5! × h0! k 1i 10635 := (− h1 k 0!i + 6!) × 3 × 5 = 5 × 3 × (6! − h0! k 1i) √ √ 14379 := (−1 − h4! k 3!i + 7!) × 9 = 9 × (7! − 3! − h4! k 1i) √ √ 15079 := − h(−1 + 5) k 0!i + 7! × 9 = 9 × 7! − h(−0! + 5) k 1i D√

E D√ E −1 + 5 k 1 × (6! + 2) = (2 + 6!) × −1 + 5 k 1 D√ E D √ E 15696 := (−1 + 5)! × 6! − 9 !k6 = − 6k 9 ! + 6! × (5 − 1)! √ 17324 := h4! k (−2 + 3!)i × 71 = 1 + 7! × h(3! − 2)! k 4i 17346 := h(−1 + 7) k 3!i + 4! × 6! = 6! × 4! + h3! k (7 − 1)i 15162 :=

q√ D E √ E D √ E √ 19844 := 4! k 4 × 8 k 9 − 1 = 1 + 98 × 4! k 4 D

20147 := 7! × 4 − h1 k 0!i − 2 D √ E 23593 := 3 k 9 × (−5 + 3!!) − 2

= −2 − h0! k 1i + 4 × 7! D√ E = −2 + (3!! − 5) × 9k3

24964 := (4 + h(6 + 9) k 4))2 D√ E √9 29789 := 9 k (8 − 7) −2

= (2 + h(4! − 9) k 6)) 4 D√ E √9 = −2 + 9 k (−7 + 8)

√

√ E√9 D√ √ E3 29793 := 3 k − 9 !+7 +2 = 2+ 9 k 7− 9 ! D

30172 := −2 + (7! − h1 k 0!i) × 3! √ 30174 := 4 × (7! − h1 k 0!i) × 3 D√ E 30282 := 2 × 8 k 2 × (0! + 3!!)

= 3! × (− h0! k 1i + 7!) − 2 √ = −3 × (h0! k 1i − 7!) × 4 D√ E = (3!! + 0!) × 2×8 k 2

30606 := ((6 + 0!)! + h6 k 0!i) × 3!

= ((3! + 0!)! + h6 k 0!i) × 6 7


Inder J. Taneja

30636 := (h6 k 3!i + (6 + 0!)!) × 3! = ((3! + 0!)! + h6 k 3!i) × 6 D E √9 E3 D√ √ √ 30897 := 3 k 0! + 8 9 k 8 + 0! − 7! = −7! + D D √ E √ E 34398 := 3! k 4 − 3!! × 9 + 8! = 8! + 9 × 3! k 4 − 3!! 34704 := 3! × h4! k (7 × 0)!i × 4! 39105 := (3!! − 9) × h1 k 0!i × 5

= h4! k (0 × 7)!i × 4! × 3! = 5 × h0! k 1i × (−9 + 3!!)

√ 39304 := 3 − h3 k 0!i × 4 = (h4 k 03i − 9)3 D √ E √ 39498 := − h3 × 9 k 4i × 9 + 8! = 8! − 9 k 4 × 9 + 3! D D √ E √ E = 8! − 5 + 6 k 9 − 3!! 39658 := 3! k 9 − 6! − 5 + 8! D √ E D√ E 39786 := − 3! k 9 ! + 7! × 8 − 6 = −6 + 8 × 7! − 9 ! k 3! D√ E D √ E √ √ 9 ! + 7! × 8 − 9 = − 9 + 8 × 7! − 9 ! k 3! 39789 := − 3! k

39792 := 39879 := 39918 := 39942 := 39948 :=

9

√ E √ − 3! k 9 ! + 7! × 9 !+2 D √ √ E 3! k 9 × −87 + 9 !! D√ √ E −3! × 9 !k 9 ! + 1 + 8! D √ E √ 9 ! + (4 × 2) ! − 3! k 9 × D √ √ E − 3 × 9 k 9 × 4 + 8!

D

D√ E = 2 × 7! − 9 ! k 3! √ D√ E = 9 + 7!/8 × 9 !k3 D√ √ E = 8! − 1 + 9 !k 9 ! × 3! D√ √ E = (2 × 4) ! − 9 ! k 9 × 3! √

9

= 8! − 4 × h9 k (9/3)i

√ √ 40058 := − 4 × (h0! k 0!i + 5!) + 8! = 8! − (5! + h0! k 0!i) × 4 D √ 40078 := − h4! k 0!i − (0 × 7)! + 8! = 8! − (−7 + h0! k 0!i)! k 4 DD√ E E D √ E 4 k 0! k 1 − 0! + 8! = 8! − h(0! + 1) k 0!i k 4 40108 := − D √ E 40158 := − h4 k 0!i − 1 − 5! + 8! = 8! − (5 + h1 k 0!i) k 4 D √ E √ 40238 := − h4 k 0!i × −2 + 3! + 8! = 8! − (3! + 2) k 04 D√

E

4 + 0! k 3 + (5 + 3)! D E √ 40362 := 4 k 0! + 3 + (6 + 2)! √ 40364 := h4 k 0!i + 3 + 64 ! D√ E 40378 := 4 k 0! + 37 + 8! 40353 :=

40479 := h4! k 0!i × 4! × 7 − 9

√ = (3 + 5)! + h3 k 0!i + 4 D √ E = (2 + 6)! + (3 + 0!) k 4 √ = 4 + 6 ! + h(3 + 0!) k 4i = 8! − 7 + h3! k 0!i + 4 = −9 + 7 × h4! k 0!i × 4!

8


Inder J. Taneja

E √ √ √ 4 k 0! × 6 × 9 + 8! = 8! + 9 ! × h6 k 0!i + 4 q √ D√ √ E √ √ 0! + 9 + 9 !! + 8! = 8! + 9 !! − 9 + 0! k 4 40998 := − 4 k

40698 :=

D√

43533 := (−4! + 3!! − 5) × h3! k 3i D√ E 43913 := −4 − 3 + 9 ! k 1 × 3!! D√ E √ √ 9 !k1 − 9 43919 := 4 + 3!! ×

= h3! k 3i × (−5 + 3!! − 4!) √ = h3! k 1i × 9 !! − 3 − 4 D√ E √ = 9 !k1 × 9 !! + 3 − 4

44164 := h(4!/4) k 1i × (6! + 4) = h(4! − 6) k 1i × h4! k 4i D √ D √ E √ E 44398 := − 4! k 4 − 3!! + 9!/8 = 8! + 9 ! × 3!! − 4! k 4 √ 44469 := h4! k 4i /4 × (6! + 9) = 96 × h4! k 4i /4 D D √ E √ √ E = −4 + 3!! × 6 k 4 − 4 44634 := −4!/4 + 6! × 3! k 4 44636 := −4 + (−4 + h6 k 3!i) × 6! = 6! × h3! k (6 − 4)i − 4 D √ E √ E √ 44642 := (4!/4)! × 6 k 4 + 2 = (2 + 4) ! × 6 k 4 + 4 D D √ E √ E √ 44646 := (4!/4)! × 6 k 4 + 6 = 6 k 4 × 6! + 4 + 4 D√ √ E D√ √ E 9 !k 9 = 9 ! k 9 × (6! + 5) + 4! 45699 := 4! + (5 + 6!) × √ 46344 := 46 × 3!! + h4! k 4!i = h4! k 4!i + 3!! × 64 D

√ E 46693 := 4 + 6 + 9k3 = 3 k 9 + 66 + 4 √ √ 46944 := 4! + 6( 9)! + h4! k 4!i = h4! k 4!i + 9 !6 + 4! D√ E D √ E 9 ! × (6! − 8) = (−8 + 6!) × 9 ! k 6 − 4! 46968 := −4! + 6 k 6

D√

E

D

47424 := h4! k 7i × 4! × 2 × 4 = 4 × 2 × h4! k 7i × 4! √ √ 49236 := 4! + 9 !! + 2 × h3! k 6i = h6 k 3!i × 2 + 9 !! + 4! E2 D √ E √4 √ 49923 := 4× 9 k 9 ×3 = 3× 2× 9 ! k9 √ √ D D √ E 4 √ E −4+8 58444 := −5! + (8 − 4)! k 4 = 4! k 4 − 5! D √ E √ 64944 := (6 × 4) k 9 ! × h4! k 4!i = 4! × 4 + 9 × h4! k 6i D E D √ E √ = (4 + 5) k 9 × 6! − 6 66954 := −6 + 6! × 9 k 5 + 4 D

D D √ E √ E 66996 := 6 × 6 + 9 k 9 × 6! = 6! × 9 k 9 + 6 × 6 √ D D √ E √ √ E 71999 := 7! − 1 + 9 !! × 9 k 9 = 9 !! × 9 k 9 − 1 + 7! 77448 := 7 × (7! + h4! k 4!i) + 8! = 8! + (h4! k 4!i + 7!) × 7 D D √ E √ E 79989 := −7 × 9 k 9 + 8! + 9! = 9! − 8! + 9 k 9 × 7 9

Inder J. Taneja


80728 := h8 k 0!i + 7 + 2 × 8!

= 8! × 2 + h7 + 0! k 8i √ 80804 := (h8 k 0!i + 8! + 0!) × 4 = 4 × (0! + h8 k 0!i + 8!) D D √ E √ E 85344 := 4 k 4 + 3!! × (5! − 8) = (−8 + 5!) × 3!! + 4 k 4 D √ D √ √ E √ E 9 !! + 3! k 9 × 5! − 5 = −5 + 5! × 9 !! + 3! k 9 93955 := D√ E D √ E 96624 := 9 ! k 6 × (6! × 2 + 4!) = (4! + 2 × 6!) × 6 k 9 ! √ √ √ √ 98404 := 9 + 8! + h4! k 0!i 4 = h4! k 0!i 4 + 8! + 9 2.2.2

√

Digit’s Order

Below are selfie numbers written in digit’s order. The numbers appearing in subsection 2.2.1 are written again except those given in section on sequential representations 2.1. √ √ E6/ 9 3969 := 3! k 9

305 := h3! k 0!i × 5 D √ E 396 := 3! k 9 ! ×6

D

4079 := − h4! k 0!i + 7! −

473 := − h4! k 7i + 3!! D √ E 492 := 4! k 9 ! ×2 D E √ 793 := 7 k 9 + 3!! D √ E 9 ! + 6! 796 := 7 k

√ 9 !!

4332 := −4! + h3! k 3!i2 D √ E 9 ! × 20 4920 := 4! k E D √ 9 ! k 3 + 7! 4937 := − 4 + D √ E 4997 := − 4 k 9/ 9 + 7!

D √ E 1446 := −1 + 4! k 4 × 6

5040 := 5! × (0! + h4 k 0!i) √ 5091 := h5 k 0!i + 9 !+1 ! √ 5097 := h5 k 0!i + 9 ! + 7!

1476 := (−1 + h4! k 7i) × 6 √ 1489 := 1 + h4! k 8i × 9 !

5640 := 5! × (6 + h4 k 0!i) D √ E 9984 := − 9 k 9 ! + 8!/4

1493 := −1 + h4! k 9i × 3! D√ E 1495 := (−1 + 4!) × 9 !k5 1920 := (−1 + 9)!/ h2 k 0!i D E √ 1968 := 1+ 9 ! k 6 ×8 D √ E 2904 := 2 × 9 ! k 0! × 4!

10285 := h1 k 0!i2 × 85 √ D √ E 04 10404 := 10 k 4

3050 := h3! k 0!i × 50 3535 := 3!! × 5 − h3! k 5i

10582 := h1 k 0!i × (5! × 8 + 2) 10584 := h1 k 0!i × 5! × 8 + 4! 10635 := (− h1 k 0!i + 6!) × 3 × 5

√

3844 := h3! k (8/4)i 4 D√ E 3905 := 3!! + 9 ! k 0! × 5 D √ E 3955 := 3 k 9 × 5! − 5 D √ E 3960 := 3 k 9 × (6 − 0!)!

10648 := h1 k 0!i

√ 6/ 4

×8

10789 := − h1 k 0!i + (7 + 8) × √ D √ E 4 12544 := (1 + 2 × 5) k 4 14056 := h(1 + 4!) k 0!i × 56

10

√ 9 !!

Inder J. Taneja


D√ E 9 !k1 √ := (−1 − h4! k 3!i + 7!) × 9 := h(−1 + 4!) k 6i × h3! k 2i := (−1 + h4! k 7i) × 60 := 1 − h4! k 8i + 7! × 3 √ := − h(−1 + 5) k 0!i + 7! × 9

14091 := h(−1 + 4!) k 0!i × 14379 14632 14760 14873 15079

15129 := h(−1 + 5) k 1i2 × 9 D√ E 15141 := (1 + (5 + 1)!) × 4k1 E D√ −1 + 5 k 1 × (6! + 2) 15162 := 15367 := (1 + 5!) × h(3! + 6) k 7i 15488 := (1 + 5!) × h(4 + 8) k 8i 15535 := (−1 + 5! + 5!) × h3! k 5i D√ E 15696 := (−1 + 5)! × 6! − 9 !k6 15851 := (1 + 5!) × h(8 + 5) k 1i E D√ 9 !+7 k 2 15972 := (1 + 5!) × q√ D√ E 8 16896 := 16 × 9 !k6

√ 17039 := −1 + h7 k 0!i × 3!!/ 9 √ 17040 := 1 + 7! × (−0! + h4! k 0!i) √ 17324 := 1 + 7! × h(3! − 2)! k 4i √ 17329 := 1 + h7 k 3!i2 × 9 17343 := h(−1 + 7) k 3i + 4! × 3!! 17346 := h(−1 + 7) k 3!h + 4! × 6! √

17424 := h(17 − 4) k 2i 4 √ 17463 := 1 + 7! × h4! k 6i − 3 √ √ 17464 := 1 + 7! × h4! k 6i − 4 √ 17466 := 1 + 7! × h(4 × 6) k 6i

√ 1 + 7! × h4! k 6i + 9 D √ E √ : = − 1 + 7 k 4 × 95 D√ E := 1 × 7!/6 × 4 k 0! D√ E := 1 + 7!/6 × 4k1 √ 1 + 8 !! + h5 k 0!i × 4! :=

17469 := 17495 17640 17641 18504

p

18 + 7! × h4! k 4!i D√ E2 := 1 + 9 !k2 ×5 D E √ := 1 + 9 ! k 6 × 80 q√ D √ E := 1 + 98 × 4! k 4 q√ := 1 × 98 × h4! k 5i

18744 :=

19225 19680 19844 19845

20147 := −2 − h0! k 1i + 4 × 7! 20182 := h2 k 0!i + 1 + 8!/2 20349 := h2 k 0!i × (3!! + h4! k 9i) √ 20880 := (h2 k 0!i + 8) × 8 + 0! !! √ √ 9 20979 := − h2 k 0!i + 7! × 9 !

E √ 23042 := 2 + 3!! × 0! + 4 k 2 D√ E 23392 := 2 + 33! × 9k2 E D√ 9k3 23593 := −2 + (3!! − 5) × D

24288 := 2 × 4!!/ h2 k (8/8)i! D √ E 24957 := − 24 k 9 + 5 × 7!

√

24964 := (2 + h(4! − 9) k 6i) 4 25048 := h2 × 5 k 0!i × h4! k 8) 25149 := h2 × 5 k 1i × h4! k 9) 25344 := 25 × 3 × h4! k 4!i 25405 := ((2 + 5)! + h4 k 0!i) × 5

√

25407 := 2 + 5 × (h4 k 0!i + 7!) 25575 := (h(2 + 5) k 5i + 7!) × 5 √ 26136 := h(2 × 6) k 1i × 3!6 27362 := 2 + h7 k 3!i × 6!/2 √ 29040 := 2 + 9 ! × (0! + h4! k 0!i) 11

Inder J. Taneja


√ 9 ! − 0! ! × h4! k 2i √ √ 9 !! − 4 × h4 k 0!i := 2 + D √ E := 29 + 5! × 4! k 9 ! √ := 2 + 9 ! × (6 + h4! k 0!i) E √9 D√ 9 k (−7 + 8) : = −2 + √ E3 D√ 9 k 7− 9 ! := 2 +

29042 := 2 + 29440 29549 29640 29789 29793

29997 := −

D

−2 +

√ √ E √ 9 ! !k 9 + 9 ! × 7!

30172 := 3! × (− h0! k 1i + 7!) − 2 30173 := − h3! k 0!i + (−1 + 7!) × 3! √ 30174 := −3 × (h0! k 1i − 7!) × 4 √ 30179 := − h3! k 0!i + 1 × 7! × 9 ! 30264 := h(3 + 0!) k 2i × 6! + 4! 30273 := h3 k 0!i + 2 + 7! × 3! D√ E 30282 := (3!! + 0!) × 2×8 k 2 30307 := h3! k 0!i + 3! × (0! + 7!) 30337 30500 30576 30606 30636

:= := := := :=

h3! k 0!i + 3! × (3! + 7!) h3! k 0!i × 500 (h3! k 0!i − 5 + 7!) × 6 ((3! + 0!)! + h6 k 0!i) × 6 ((3! + 0!)! + h6 k 3!i) × 6

30667 := (h3! k 0!i + 6! × 6) × 7 √ 6 30866 := h3! k 0!i × 8 −6 √ E 9 D √ 30897 := 3 k 0! + 8 − 7!

33589 := (− h3! k 3!i + 5 × 8!) / D √ E 3! 33597 := 3 × 5 k 9 − 7!

√ 9 !

33696 := 3!3 × h(6 + 9) k 6i D √ E 34398 := 3! k 4 − 3!! × 9 + 8! E D√ 9 !k5 34495 := 3!! × (4! + 4!) −

√ 9 !! + 6! 34496 := − h3! k 4i + 4! × 34624 := h3! k 4i + 6! × 2 × 4! 34702 := (3!! × 4! + h7 k 0!i) × 2 34704 := 3! × h4! k (7 × 0)!i × 4! D √ E 34937 := 3! + 4 k 9 × (3!! − 7)

√ E √ 34944 := 3! × 4! k 9 − 4 × 4! D√ E2 34972 := 3 + 4×9 k 7 E √9 D√ 9k3 35939 := −3 + 5 + D E √4 √ 9 !k6 35964 := (3 + 5)! −

D

36567 := (−3 + 6!) × h5 k (−6 + 7)i

37560 := 3! × 7! + 5! × h6 k 0!i 38073 := h(3 + 8) k 0!i × 73 38520 := 3!! + 8! − 5! × h2 k 0!i D√ E 39050 := 3!! + 9 ! k 0! × 50 D√ E 39052 := 3!! + 9 k 0! × 52

2

31232 := h3! k 1i × 23

32830 := h3! k 2i + 8(3!−0!) 32854 := h3! k 2i + 85 + 4! 33044 := (3!! + h3 k 0!i) × 44 33124 := h(3 × 3) k 1i2 × 4 33327 := (h3! k 3i + 3!)2 × 7 D √ E2 33492 := 3 + (−3! + 4!) k 9

39105 := (3!! − 9) × h1 k 0!i × 5 D√ E √4 39204 := 3! × 9 k 2 + 0! 39228 := −3! × h(9 × 2) k 2i + 8! D √ E 9 39284 := 3 × 2 − 8 k 4 √ 39304 := 39 − h3 k 0!i × 4 12

Inder J. Taneja


39328 := −3!! − h(9 × 3) k 2i + 8! 39378 := −3! × h(9 + 3!) k 7i + 8! E √ D√ 39435 := 3!! − 9 × 4+3 k 5 39438 := − h(3 × 9) k 4!i × 3 + 8! D √ E 39468 := − 3 k 9 × 4 − 6! + 8! D √ E 39484 := −3!! − 9 k 4 + 8! − 4! √ 39498 := − h(3 × 9) k 4i × 9 + 8! D E √ 39508 := −3!! − 9 k 5 − 0! + 8! √ 39535 := −3!! + 9 + 5 ! − h3! k 5i D E √ 39538 := − 3! k 9 − 5 − 3!! + 8!

√ E 39658 := 3! k 9 − 6! − 5 + 8! D √ E 39690 := 3! k 9 × (6! − 90) E D √ 39738 := − 3 × 9 k 7 × 3! + 8! D √ E 39768 := − 3! k 9 + 7! − 6 × 8 D √ E 39786 := − 3! k 9 ! + 7! × 8 − 6 D √ E √ 39789 := − 3! k 9 ! + 7! × 8 − 9 D

√ D √ E 39792 := − 3! k 9 ! + 7! × 9 !+2

40078 := − h4! k 0!i − (0 × 7)! + 8! √ 40079 := − h4! k 0!i + 0! + 7 × 9 ! 40098 := − h(4! − 0!) k 0!i + 9 + 8! E E DD√ 4 k 0! k 1 − 0! + 8! 40108 := − DD√ E E 40109 := − 4 k 0! k 1 + (−0! + 9)! 40158 := − h4 k 0!i − 1 − 5! + 8! 40185 := −4 − h0! k 1i + 8! + 5! √ 40238 := − h4 k 0!i × −2 + 3! + 8! D E 40247 := h4! k 0!i × 24 k 7 40279 := − h4 k 0!i + (2 + 7)!/9 40281 := − h4 k 0!i + 2 + 8! × 1 D√ E 40299 := − 4 k (0 × 2)! + 9!/9 E D√ √ 4 k 0! + 3! + 4 ! − 0! 40340 := D√ E 40341 := 4 k 0! + (3 + 4 + 1)! D√ E 40353 := 4 + 0! k 3 + (5 + 3)! 40358 := h4 k 03i − 5 + 8! 40361 := h4 k 0!i + (3 + 6 − 1)! D E √ 40362 := 4 k 0! + 3 + (6 + 2)! √ 64 ! 40364 := h4 k 0!i + 3 + E √ D√ 4 k 0! + 36 + 8! 40368 := 40378 :=

D √ √ E 39879 := 3! k 9 × −87 + 9 !! √ √ D √ E 39894 := − 3! k 9 ! + 8! − 9 !!/ 4 D√ √ E 39918 := −3! × 9 !k 9 ! + 1 + 8!

√ E √ 39942 := − 3! k 9 × 9 ! + (4 × 2) ! D √ √ E 39948 := − 3 × 9 k 9 × 4 + 8! √ 40058 := − 4 × (h0! k 0!i + 5!) + 8! 40068 := − (h4 k 0!i + 0!) × 6 + 8! 40073 := − h4! k 0!i + (0! + 7)! − 3! D

D√

E 4 k 0! + 37 + 8!

40428 := h(4 + 0!) k 4i × 2 + 8! D√ √ E 40448 := 4 + 0! k 4 × 4 + 8! 40479 := h4! k 0!i × 4! × 7 − 9 40482 := h4 k 0!i × 4 + 8! − 2 40485 := h4 k 0!i + 4 + 8! + 5! DD√ E E 40535 := 4 k 0! k 5 + (3 + 5)! DD√ E E 40538 := 4 k 0! k 5 + 3 + 8! √ 40564 := h4! k (−0! + 5)i + 64 ! D E √ 40571 := 4! k 0! + 5! + (7 + 1)! 13

Inder J. Taneja


D E √ 40578 := 4! k 0! + 5! + 7 + 8! E E DD√ 4 k 0! + 5 k 8 + 8! 40588 := √ 40648 := h4 k 0!i × 64 + 8! 40685 := h4! k (−0! + 6)i + 8! + 5! D√ E √ 40698 := 4 k 0! × 6 × 9 + 8! 40783 := − h(4! + 0!) k 7i + 8! + 3!! D √ √ E 4 + 0! !! − 9 k 4 + 8! 40948 := 40959 := h4 k 0!i × (−9 + 5!) × 9 q √ √ 40998 := − 4 k 0! + 9 + 9 !! + 8! √ 40999 := − h4 k 0!i + 9 !! + 9!/9 := h(4! − 1) k 2i × 4 + 8! := h4! k (1 + 2)i + 8! + 3!! := h4! k (1 + 2)!i + 8! + 6! := 4! × h(1 + 3) k 2i + 8! E D√ √ 4 k 1 + 4! 9 × 3 41493 :=

41248 41283 41286 41328

41637 :=

D√

E 4 k 1 + 63! − 7!

41868 := h4! k 18i × 6 + 8! √ 41934 := h4! k 1i × − 9 ! + 3!!/4 q D √ E 41984 := (4 × 1)9 × 8 k 4 42088 := h(4! − 2) k 0!i × 8 + 8! 42498 43152 43248 43404 43533

:= h4! k (−2 + 4)i × 9 + 8! := (−4! + 3!!) × h(1 + 5) k 2i := h(4 × 3) k 2i × 4! + 8! := 4! + 3!! × h4! k 0!i /4 := (−4! + 3!! − 5) × h3! k 3i

43555 := (h4! k 3i + 5!) × 5! − 5 43560 := (h4! k 3i + 5!) × (6 − 0!)! D√ E 43591 := −4! + (3!! − 5) × 9 !k1 43896 := −4! + h3! k (−8 + 9)i × 6! D√ E 43908 := −4 + 3!! × 9 ! k 0! − 8

D√ E √ 43909 := − 4 + 3!! × 9 ! k 0! − 9 E D√ 9 ! k 1 × 3!! 43913 := −4 − 3 + D√ E √ 43914 := −4 + 3!! × 9 !k1 − 4 D√ E √ 4×3 !× 9 ! k 1 −5 43915 := 43916 := −4 + h(−3 + 9) k 1i × 6! D√ E 43917 := 4 + 3!! × 9! k 1 − 7 E D√ √ 8 43918 := − 4 + 3!! × 9 !k1 D√ E √ √ 43919 := 4 + 3!! × 9 !k1 − 9 √ D√ E 43920 := 4×3 !× 9 × 2 k 0! D√ E √ 43922 := 4 + 3!! × 9 ! k (2/2)

D√ √ E 9 ! k 2/ 4 43924 := 4 + 3!! × √ 43930 := 4 + 3! + 9 !! × h3! k 0!i D√ E 43944 := 4! + 3!! × 9 ! k (4/4) 44164 := h(4!/4) k 1i × (6! + 4) √ D√ E 44392 := −4 + 4×3 ! × 9 !k2

D√ √ E 4 + (−4 + 3!!) × 9 !k 4 D E √ 44398 := − 4! k 4 − 3!! + 9!/8 44394 :=

√

44469 := h4! k 4i /4 × (6! + 9) √ 44590 := 4 × h4! k 5i × h9 k 0!i 44632 := −4 − 4 + 6! × h3! k 2i D √ E 44634 := −4!/4 + 6! × 3! k 4 44636 := −4 + (−4 + h6 k 3!i) × 6! 45336 := (−4 + 5! × h3! k 3i) × 6 √ D √ E 4 45479 := − 4 + 5 × 7 k 9 D √ E 45664 := 45 + 6! × 6 k 4 14


Inder J. Taneja

45699 := 4! + (5 + 6!) ×

D√ √ E 9 !k 9

45835 := − h4! k 5i − 8! + 3!! × 5! D√ E √ 4+5 !+ 9 ! k 0! × 9 45909 := 46056 := −4! + h6 k (−0! + 5)i × 6! 46326 := h4! k 6i + 3!! × 26

46 × 3!! + h4! k 4!i √ 46670 := 4 − 6! × (6 − h7 k 0!i) 46344 :=

46763 := h(4 + 6) k 7i + 63! √ 9 !(0!+2)! 46902 := h4! k 6i + √ D√ E 46930 := 4 + 6! × 9 ! k (3! − 0!)

√

46944 := 4! + 6( 9)! + h4! k 4!i √ D√ E 46954 := 4 + 6! × 9 ! k 5 + 4! D √ E 46968 := −4! + 6 k 9 ! × (6! − 8) 47424 := h4! k 7i × 4! × 2 × 4 47488 := (h4! k 7i × 4! + 8) × 8

49200 49236 49248 49704

53824 := h(5 × 3 + 8) k 2i

4

√

54294 := 5 + (h4! k 2i − 9) 4 √ 54590 := −5 × 4 − 5! × h9 k 0!i √ 54739 := −5 + 4! + h7 k 3!i × 9 !! 55080 := 5! × h5 k 0!i × (8 + 0!)

√

47496

√

√ E := −4! + (7 − 4)! k 9 ! × 6! D √ E 9 ! × 200 := 4! k √ := 4! + 9 !! + 2 × h3! k 6i DD E √ E := 4! × 9 × 2 k 4 k 8 √ √ := 4! − 9 !! × − h7 k 0!i + 4 D

58035 := 5! + h8 k 0!i × (3!! − 5) 58320 := (−5 + 8)!! × h(3! + 2) k 0!i √ D √ E 4 58444 := −5! + (8 − 4)! k 4 √ 58920 := 5! × 8 9 − h2 k 0!i D √ E 9 ! × (6! + 8) 58968 := −5 + 8 k √ p 59009 := −5! + 9h0!k0!i / 9 59045 := h(−5 + 9)! k 0!i × h4! k 5i D E √ √ 9 !! × 8 k 8 + 0! 59880 := 5! + D √ √ E 59926 := 5 + 9 k 9 × (2 + 6!) 60846 := (h6 k 0!i + 8!/4) × 6 61832 := (6! − 1) × h8 k 3!i − 2 D √ E 61834 := (6! − 1) × 8 k 3 × 4 61920 := 6! × h(−1 + 9) k (2 + 0!)!i 62496 := h6 × 2 k 4i × 9!/6! 63084 := (6! + h3 k 0!i) × 84 63360 := 6! × (−3 + h(3 + 6) k 0!i) D E √ 63888 := (6! + 3!) × 8 k 8 × 8 √ 64080 := 6! × − 4 + h(0! + 8) k 0!i √ 2 64729 := (6 + h4! k 7i) + 9 !! D √ E 64944 := 6 × 4 k 9 ! × h4! k 4!i

√ D√ E 49790 := 4 + 9 ! × 7! − 9 ! k 0! √ 64980 : = 6! + 4 × (9 + h8 k 0!i) D E2 √ D E √ 49923 := (4 × 9) k 9 × 3 66954 : = − 6 + 6! × 9 k 5 + 4 D√ √ E3 D √ E 49933 := 4 + 9k 9 − 3!! 66996 := 6 × 6 + 9 k 9 × 6!

√

51984 := (5! + h(1 + 9) k 8i) 3

53347 := −5 + 3! × h4! k 7i

4

68085 := (6! + h8 k 0!i) × 85 68400 := 6! × (84 + h0! k 0!i) 15


Inder J. Taneja

D√ E 69092 := 6! + 9 k 0! × 92 √ 69336 := 6! × h9 k 3!i + 3!6 √ 69408 := 6 × 9 ! × h4! k 0!i × 8 D √ E 69435 := 6 k 9 + 4!3 × 5 E √ D√ 7 69792 := −6 + 9 × 9k2

71999 := 7! − 1 +

√ D √ E 9 !! × 9 k 9

72595 := (h(7 + 2) k 5i + 9!) /5 D √ E 73435 := (7 + 3) k 4 × 3!! − 5 D √ E 73439 := (−7 + 3!!) × (4 + 3!) k 9 D √ E 73464 := (7 + 3) k 4 × 6! + 4!

D√ E √ 90659 := − 9 ! k 0! + 6! × 5! + 9 ! D√ E √ 9 !! + 0! + 7! × 4k1 90741 := − D√ E 90782 := 9 k 0! + 7! + 8! × 2 √ 9 !! + 0! × h(8 + 4) k 6i 90846 := E2 D√ 9 k 1 × 95 91295 :=

D√ E2 9 ! k 2 × 4! − 8 √ D √ E 3! 93259 := 9 ! ×2− 5 k 9 √ √ 9 !! × h3! k 5i − 8 × 4 93584 :=

92248 :=

93852 := (−9 + 3!!) × h(8 + 5) k 2i √ D √ E 93955 := 9 !! + 3! k 9 × 5! − 5

74347 := 7 × 43 × h4! k 7i D √ E 74949 := 7 × 4 k 9 × h4! k 9i 77448 := 7 × (7! + h4! k 4!i) + 8! D √ E 79345 := 7 × 9!/ 3 k 4 − 5 √

79524 := h7 × (9 − 5) k 2i

93984 94080

4

√ E 79989 := −7 × 9 k 9 + 8! + 9! √ 80482 := − h8 k 0!i + 4 + 8! × 2

94089

D

80728 := h8 k 0!i + 7 + 2 × 8! 80802 := (h8 k 0!i + 8!) × 02 80804 := (h8 k 0!i + 8! + 0!) ×

D √ √ E 9 !! + 3! k 9 × (6 − 0!)! D√ E √ := 9k3 × 9 !! − 8 × 4 D√ E := 9! × 4 k 0! / h8 k 0!i D√ E := 9 + 4 k 0! × 8!/9 √ D E 2 := 9 !! × 4 + 3 k 0! D√ E := 9 ! k 6 × (6! × 2 + 4!)

93960 :=

√ 4

80832 := (8! + h(0! + 8) k 3!i) × 2 D √ E 85344 := (−8 + 5!) × 3!! + 4 k 4 86400 := (8! − 6!) × 4!/ h0! k 0!i D√ E √ 89264 := −8 + 9 ! k 2 × 6! × 4 √ √ 89304 := 8 × 9 × h3! k 0!i 4

94320 96624

√ D √ E 96795 := − 9 + 6! × 7 + 9 !k5 √ √4 97344 := 9 × h(7 + 3) k 4i √ D √ E 97920 := 9 !! × 7 + 9 ! k (2 + 0!)! √ √ 98404 := 9 + 8! + h4! k 0!i 4 D√ √ E2 99225 := 9 ! k 9 × 25 √ 99360 := 9! − 9 ! × 3!! × h6 k 0!i

16

Inder J. Taneja

2.2.3


Reverse Order of Digits

Below are selfie numbers written in reverse order of digits. The numbers appearing in subsection [?] are written again except those given in subsection on sequential representations 2.1. D√ E 0328 := 8 × h(−2 + 3!) k 0!i 147 := 7 × 4k1 √ 9 !!/2 − h3 k 0!i 0329 : = 241 := −1 + h4! k 2i D √ E 0357 := 7 × h5 k (3 × 0)!i 244 := 4! k 42 0362 := 2 × h(6 × 3) k 0!i 264 := h4! k (6 − 2)!i D √ E 0364 := 4 × h(6 + 3) k 0!i 396 := 6 k 9 ! × 3! 0105 := 5 × h(0! + 1) k 0!i √ 0109 := 9 ! − 0! ! − h1 k 0!i 0122 := 2 × h(2 + 1)! k 0!i 0124 := 4 × h(2 + 1) k 0!i 0132 := 2 × 3! × h1 k 0!i 0182 := 2 × h(8 + 1) k 0!i D√ E 0183 := 3 × 8 + 1 ! k 0! D√ E 0186 := 6 × 8 + 1 k 0! 0201 := h10 × 2 k 0!i 0217 0231 0241 0243

:= 7 × h(1 + 2) k 0!i := h(−1 + (3! − 2)!) k 0!i := h1 × 4! k (2 × 0)!i := h(3! × 4) k 2i + 0!

√ 9 ! + h4! k 2i + 0! 0249 := D E 0251 := (1 × 52 ) k 0! D√ 16 ! k h2 k 0!i 0261 :=

0366 0375 0408 0427

: = 6 × h6 k (3 × 0) ! i := 5 × (h7 k 3!) − 0!i := 8 × h(0! + 4) k 0!i := 7 × h(2 + 4) k 0!i

√ := 1 + 5! × h4 k 0!i := 5 × h(5 + 4) k 0!i : = 9 × h5 k (4 × 0) ! i := 2 × h(8 − 4)! k 0!i D E √ 0483 := 3 × 8 × 4 k 0!

0451 0455 0459 0482

0486 := 6 × h8 k (4 × 0)!i D E √ 0488 := 8 × 8 − 4 k 0! √ 0492 := 2 × 9 ! × h4 k 0!i D E √ 0497 := 7 × 9 − 4 k 0! D √ E √ 0498 := 8 k 9 × 4 + 0! !

0273 := 3 × h(7 + 2) k 0!i 0279 := 9 ×

D√

7 + 2 k 0!

E

0287 := 7 × h(8/2) k 0!i 0305 := h5 k 0!i × 3! − 0! 0306 := 6 × h(−0! + 3!) k 0!i D E 0324 := 4 × 23 k 0!

0504 :=

D√

E 4 k 0! × (5 − 0!)!

0546 := 6 × h(4 + 5) k 0!i D√ E 0549 := 9 × 4 + 5 ! k 0! √ 0589 := 9 !! − h(8 + 5) k 0!i 0637 := 7 × h(3 + 6) k 0!i 17


Inder J. Taneja

D√ E 0648 := 8 × 4 + 6 k 0! √ 9 ! × 5! − h6 k 0!i 0659 := √ 0709 := 9 !! − h0! k (7 × 0)!i D √ E 0742 := 2 k 4 + (7 − 0!)!

4794 4973 4974 4976

0783 := 3!! − 8 + h7 k 0!i 0791 :=

√

4979

1 × 9 !! + h7 k 0!i

0819 := 9 × h(1 + 8) k 0!i 0847 := 7 × h(4 + 8) k 0!i E D √ 9 ! k 0! 0968 := 8 × 6 + D √ E 0987 := 7 × 8 + 9 ! k 0! D E 1255 := 5 × 52 k 1 1288 := 8 × h(8 × 2) k 1i D √ E 1342 := 2 k 4 × h3! k 1i 1359 := 9 × h(5 × 3) k 1i 1403 := h3! k 0!i × (4! − 1) := −8 + 3! × h4! k 1i := 9 × h(4 × 4) k 1i := 5 × h(5 + 4!) k 1i := h4! k 9i × (4 − 1)! D √ E 1928 := 8 × −2 + 9 ! !k1 1438 1449 1455 1494

E √ 4!/ 9 k 1 E D√ 5 9 !k1 1952 := 2 × 1944 := 4! ×

D

1984 := h4! k 8i × (9 − 1) D√ E 2 2809 := 9 ! k 0! − 8 3024 := 4! × h2 k 0!i × 3! 3044 := 4 × (h4 k 0!i + 3!!) 3509 := − h9 k 0!i + 5 × 3!! D√ E 3955 := −5 + 5! × 9k3 √

4356 := h(6!/5!) k 3!i 4 4392 := h(2 × 9) k 3i × 4!

5904

√ E q√ := − 4! k 9 ! + 74 ! D√ E := −3 + 7! − 9 !k4 D√ E √ 9 !k4 := − 4 + 7! − E D√ 9 !k4 := 6! × 7 − D√ E √ 9 !k4 := 9 + 7! − √ := h4 k 0!i × 9 !!/5 D

00123 := h(3! × 2) k 1i + 0! + 0! √ 00124 := 4 × (h(2 + 1)! k 0!i + 0!) 00129 := 9 × 2 + hh1 k 0!i k 0!i 00132 := h(2 × 3! + 1) k 0!i + 0! 00138 := (8 + h3! k 1i) × (0! + 0!) 00142 := 2 × h((4 − 1)! + 0!) k 0!i 00157 := 7 × (5 − 1)! − h0! k 0!i D√ E 00159 := 9 × 5 k 10 − 0! 00162 := 2 × h(6 + 1 + 0!) k 0!i 00168 := (8 + 6) × (h1 k 0!i + 0!)

00179 :=

√ 9 !!/ (−7 + h1 k 0!i) − 0!

00182 := h2 × (8 + 1) k 0!i + 0! 00183 := 3 × h(8 − 1 − 0!) k 0!i 00189 := 9 × h(−8 + 10) k 0!i 00192 := h∗2 × 9 + 1) k 0!i + 0! E D√ 00193 := 3! × 9 k 1 + 0! + 0! 00198 := (8 + 9 + 1) × h0! k 0!i 00213 := 3 × h(1 + 2)! + 0! k 0!i E E DD√ 4 k 1 k 2 + 0! + 0! 00214 :=

00215 00218 00219 00231 18

:= 5 − 1 + hh2 k 0!i k 0!i := 8 − 1 + hh2 k 0!i k 0!i := 9 − 1 + hh2 k 0!i k 0!i := h(1 × 3 + 20) k 0!i

Inder J. Taneja


00234 := h4! k (3 + 2)i − h0! k 0!i 00247 := 7 + h4! k (2 × 0)!i − 0! D√ E 00248 := 8 × 4 + (2 × 0)! k 0!

00476 := 6! + 7 − h(4! + 0!) k 0!i √ 00491 := −1 + 9 ! ! × 4 + h0! k 0!i

00249 := 9 + h4! k ((2 × 0)! − 0!)i D E 2 00256 := 6 + 5 k 0! − 0!

00495 := 5 × 9 × h(4 × 0)! k 0!i

00261 := h(1 × 6 + 20) k 0!i 00271 := h(1 × 7 + 20) k 0!i 00275 := 5 × (7 − 2) × h0! k 0!i

D √ E 00498 := 8 k 9 × (4 + 0! + 0!)

00281 := h(1 × 8 + 20) k 0!i 00284 := 4 × h(8 − (2 × 0)!) k 0!i 00291 := h(1 × 9 + 20) k 0!i r √ 9 !! + 2 × h0! k 0!i 00297 := 7 + 00312 := 2 × 13!/ h0! k 0!i! 00315 := 5 × (1 + h3! k (0! + 0!)i) 00341 := (1 + 4! + 3!) × h0! k 0!i

√

00395 := −5! + 9!/3!! + h0! k 0!i 00396 := 6 × (9 − 3) × h0! k 0!i 00421 := −1 + 2 ×

4 k 0! k 0!

E

00426 := 6 × h(2 + 4 + 0!) k 0!i 00437 := 7 × h3! k 4i − h0! k 0!i 00452 := 2 × (−5 + h(4! − 0!) k 0!i) D√ E 00453 := 3 × 5! + 4 + 0! k 0! 00462 := 2 × h(6 × 4 − 0!) k 0!i 00463 := −3! + 6! − h(4! + 0!) k 0!i

00543 := 3 × h(4! − 5 − 0!) k 0!i D√ E 00567 := 7 × 65 − 0! k 0!

00583 := (3! × 8 + 5) × h0! k 0!i 00594 := (49 + 5) × h0! k 0!i 00627 := −7 + 26 × h0! k 0!i

00374 := (4! + 7 + 3) × h0! k 0!i √ √ 00394 := − 4 + 9 ! × 3! × h0! k 0!i

E

:= 4!2 − h5 k 0!i − 0! := 8 × (−2 + 5)! × h0! k 0!i := (9 × 3! − 5) × h0! k 0!i := 2 × h(−4! + h5 k 0!i) k 0!i

√ D E √ 00579 := 9 !! − 7 × 5 − 0! k 0!

2 + 7 ! × (h3! k 0!i + 0!)

DD√

00524 00528 00539 00542

00568 := 8 × h(6 + (5 × 0)!) k 0!i D√ E 00573 := 3!! − 7 × 5 − 0! k 0!

00342 := 2 × h(4! − 3! − 0!) k 0!i √ 00346 := 6!/ 4 − 3 − h0! k 0!i 00349 := (9 − 4)! × 3 − h0! k 0!i 00369 := 9 × h(6 − 3 + 0!) k 0!i 00372 :=

√ 00468 := 86 − 4 × h0! k 0!i 00469 := −9 + 6! − h4! k 0!i − 0! √ 00473 := −3! + 74 × h0! k 0!i

00634 := −4! + 3!! − h6 k 0!i − 0! 00639 := 9 × h(3! + (6 × 0)!) k 0!i 00653 := − h3! k 5i + 6! − 0! − 0! √ 00654 := 4 + 5 !! − 6 × h0! k 0!i 00658 := (8 − 5)!! − h6 k 0!i − 0! q 00684 := 4! + (8! − 6!) × h0! k 0!i √ 00689 := 9 !! − h(8 − 6 + 0!) k 0!i 00695 := −5 − 9 + 6! − h0! k 0!i 00724 00731 00732 00743 19

:= (4 + 2)! − 7 + h0! k 0!i := (13 − 7)! + h0! k 0!i := 2 × 3! × h7 − 0! k 0!i := 3 × h4! k 7i + 0! + 0!

Inder J. Taneja


p

18 + 7! × h0! k 0!i √ 00789 := 9 !! + 8 + h(7 − 0!) k 0!i

01254 := (−4 + 5! − 2) × h1 k 0!i √ 01298 := 8 − 9 ! − 2 × h1 k 0!i

00791 := −1 + 9!/7! × h0! k 0!i 00792 := (2 + 9) × (h7 k 0!i + 0!) √ 00794 := 4 + 9!/7! × h0! k 0!i

01302 := h2 k 0!i × (h3! k 1i + 0!)

00781 :=

00837 := 7!/3! + 8 − h0! k 0!i 00843 := 3 × h(4 × (8 − 0!)) k 0!i E D√ 4 × (8 − 0!) k 0! 00846 := 6 × √ 9 !! + 5! + 8 + h0! k 0!i q√ := 1 × 98 × h0! k 0!i := (2 + 9) × h8 k 0!i + 0! √ : = 7 × −1 + 9 ! ! + h0! k 0!i D√ √ := 64 k 9 ! × h0! k 0!i

01337 01345 01353 01364

:= 7 × h(3 × 3! + 1) k 0!i := −5! + 4! × h3! k 1i + 0! := (−3 + 5! + 3!) × h1 k 0!i := (4! + 6!) /3! × h1 k 0!i

00859 := 00891 00892 00917 00946

D√ E 00956 := 6! − 5 + 9 + 0! ! k 0! E D√ 9 ! − 0! k 0! 00957 := 7!/5 − 00968 := (8 + 6!/9) × h0! k 0!i D E √ 00984 := 4! × 8 − 9 − 0! k 0! 01044 := 4 × h(4! + 0! + 1) k 0!i 01045 := (5! − 4! − 0!) × h1 k 0!i 01048 := 8 × h(4! − h0! k 1i) k 0!i DD√ E E 01055 := 5 × 5 − 0! k 1 k 0! 01059 := 9 × 5! − h(0! + 1) k 0!i 01069 := 9 × (6 − 0!)! − h1 k 0!i 01089 := (98 + 0!) × h1 k 0!i 01155 := 5 × h((5 − 1)! − 1) k 0!i D√ E 01197 := 7 × 9 ! + 11 k 0!

01369 :=

√ 9 !! + 6! − h(3! + 1) k 0!i

01375 := (5 + 7! × 3) / h1 k 0!i 01386 := 6 × h(8 × 3 − 1) k 0!i √ 01397 := 7 + 9 !!/3! × h1 k 0!i √ √ 9 !! + 9 !! − h(3 + 1) k 0!i 01399 :=

01407 := 7 ×

D

−0! +

D√ E E 4 k 1 k 0!

01408 := (8 + (0! + 4)!) × h1 k 0!i √ √ 01429 := 9 × 2 ! × 4 − h1 k 0!i √ 01431 := (1 + 3!!) × 4 − h1 k 0!i √ 01437 := (−7 + 3!!) × 4 + h1 k 0!i

1+0! √ E 01444 := 4 k 4 −4 √ 01445 := 5 + 4 ! × h4! k 1i − 0! D

01447 := (7 − 4)! × h4! k 1i + 0! 01448 := 8 × h(4! − (4 − 1)!) k 0!i √ 01451 := (1 + 5)! × 4 + h1 k 0!i

01205 := 5 × h(0! + 2 + 1)! k 0!i 01206 := 6 × h(−0! + 21) k 0!i 01224 := 4! × h(2 + 2 + 1) k 0!i √ 01229 := 9 !! × 2 − h21 k 0!i 01245 := 5 × h4! k (−2 + h1 k 0!i)i 01247 := 7!/4 − 2 − h1 k 0!i

01452 := (−2 + 5)! × (h4! k 1i + 0!) √ 01463 := (3! + 6!) × 4 + h1 k 0!i p 01469 := 96 × 4 + h1 k 0!i √ 01498 := −8 + 9 ! × h(4! + 1) k 0!i 20

Inder J. Taneja

01499 :=


√ 9 ! × (9 + h4! k 1i) − 0!

01506 := 6 × h(0! + (5 − 1)!) k 0!i 01525 := 52 × h(5 + 1) k 0!i E D √ 01547 := 7 × 4! − 5 − 1 k 0! 01549 := (9 + 4) × 5! − h1 k 0!i 01559 := 01575 01584 01595 01626

E D√ 9 ! k 5 × (5 − 1)! − 0!

:= 5 × 7!/ (5 + h1 k 0!i) := 4! × (8 − 5)! × h1 k 0!i := h5 + 9 k 5i × h1 k 0!i := 6 × h(26 + 1) k 0!i

√

01669 := 9!/ 66 − h1 k 0!i 01672 := 2 × 76 × h1 k 0!i 01688 := 8 × hh(8 − 6) k 1i k 0!i √ 01697 := 7!/ 9 + 6 + h1 k 0!i 01746 := 6 × h(4 × 7 + 1) k 0!i 01749 := (−9 + 4! × 7) × h1 k 0!i *q + p 01757 := 7 × 5(7+1) k 0! 01848 :=

D

E √ 8 × 4 k 8 × h1 k 0!i

√

9 × 7!/8 − h1 k 0!i D√ E 01897 := 7 × 9 × 81 k 0! 01879 :=

D√

p 02044 := −4 + 4h0!k(2×0)!i √ √ 02099 := 9 × 9 !! − h(0! + 2)! k 0!i 02149 := 9 × h4! k 1i − 20 02184 := 4! × h(8 + 1) k (2 × 0)!i √ 02196 := 6 × 9 ! × h(1 + 2)! k 0!i 02248 := 8 × h(4! + 2 + 2) k 0!i 02255 := 55 × h(2 + 2) k 0!i D E 02259 := 9 × 52 k (2 × 0)! 02283 := (3! × 8)2 − h2 k 0!i 02305 := (h5 k 0!i − 3)2 + 0! 02346 :=

D

√ E 6 k 4 + 3!! × (2 + 0!)

02349 := 9 × h(4 × 3! + 2) k 0!i 02352 := (−2 + 5! − 3!) × h2 k 0!i √ 9 !! + 7!/3 − h2 k 0!i 02379 := 02394 := ((−4 + 9)! − 3!) × h2 k 0!i 02415 02419 02436 02439 02445

:= (5! − 1 − 4) × h2 k 0!i := (9 + 1) × h4! k 2i − 0! := (6!/3! − 4) × h2 k 0!i := 9 × h(3 + 4!) k (2 × 0)!i := 5 × (h4! k 4i × 2 + 0!)

E 9 × (8 − 1) k 0! D√ E 01923 := h3! k 2i × 9 k 1 + 0! √ 01924 := h4! k 2i + 9 !! × (1 + 0!)

√ 02454 := −4! + 5! − 4 × h2 k 0!i √ 02495 := 5 × 9 !! − h(4! − 2) k 0!i E √ 02499 := 9!/ 9 ! × 4! − h2 k 0!i

01933 := 3!3 × 9 − h1 k 0!i 01942 := −2 + 4! × h(9 − 1) k 0!i

02513 := −3! − 1 + 5! × h2 k 0!i 02514 := − (4 − 1)! + 5! × h2 k 0!i

01899 := 9 ×

01947 := (7 × 4! + 9) × h1 k 0!i √ 01955 := (5! − 5) × 9 ! + h1 k 0!i D√ E 01967 := 7 × 6! + 9 k h1 k 0!i D√ E 01968 := 8 × 6 × 9 + 1 k 0! 02037 := 7 × h(h3 k 0!i − 2) k 0!i

02515 := −5 + 1 × 5! × h2 k 0!i 02532 := 2 × 3! + 5! × h2 k 0!i 02535 := 5 × 3 + 5! × h2 k 0!i 4 02541 := 1 + 5! × h2 k 0!i D √ E 02542 := 2 k 4 + 5! × h2 k 0!i 21

Inder J. Taneja


√

54 + 5! × h2 k 0!i √ 02562 := −2 + 6 + 5! × h2 k 0!i 02545 :=

02568 := 8 × 6 + 5! × h2 k 0!i r  q√ 02583 :=  38 + 5! × h2 k 0!i 02604 := (4 + (−0! + 6)!) × h2 k 0!i

02646 :=

D

E √ 6 × 4 k 6 × h2 k 0!i

√ 02651 := 1 + 5! × h(6 − 2)! k 0!i 02667 := (7 + 6!/6) × h2 k 0!i 02684 := −4 + 8!/ (−6 + h2 k 0!i) √ 02859 := 9 × 5! × 8 − h2 k 0!i

02892 := 2 ×

√ 9 ! × h(8/2)! k 0!i

02904 := 4! × h(0! + 9 + 2) k 0!i D √ E 02943 := 3 k 4 × 92 − 0! 02964 := 4 × ((−6 + 9)!! + h2 k 0!i) D√ E 02968 := 8 × 6 × 9 ! k 2 − 0!

03396 03445 03482 03503 03509

:= 6 × (9 × h3! k 3i − 0!) := 5 × ((4!/4)! − h3 k 0!i) := 2 × (8!/4! + h3! k 0!i) := h3 k 0!i × (5! − 3! − 0!) := − h9 k 0!i + 5! × 30

03552 := 25 × h(5 + 3!) k 0!i √ 03559 := − 9 ! + (5! − 5) × h3 k 0!i √

03595 := −5 9 + 5! × h3 k 0!i 03624 := 4! × h(2 × 6 + 3) k 0!i 03658 := (−8 + 5! + 6) × h3 k 0!i 03661 03696 03843 03864 03865

:= (−1 + 6) × 6! + h3! k 0!i := 6! + 96 × h3 k 0!i := h3! k (4!/8)i × h3! k 0!i := 4! × h(6 × 8/3) k 0!i := 5 × (6! − 8 + h3! k 0!i) √

09

× h3! k 0!i √ 03905 := 5 × 09 !! + h3! k 0!i √ 03906 := (6 − 0!)! + 9 ! × h3 k 0!i 03904 := 4

√

03945 := −5! + 4( 9)! − h3 k 0!i √ D√ E √ √ 02995 := 5 × 9 !! − 9 ! × 2 k 0! 03949 := 9 !4 × 9 + h3! k 0!i 03124 := 4 × ((2 + 1)!! + h3! k 0!i) 03142 := −2 + 4! × h13 k 0!i 03164 := 4 × (6! + h(1 + 3!) k 0!i) 03195 := 5 × 9 × h(1 + 3!) k 0!i √ 03264 := 46 × h(2 + 3) k 0!i √ 03294 := 4! + 9 × 2 × h3! k 0!i D√ E 03295 := 5 × − 9 ! k 2 + 3!! + 0!

03315 03343 03355 03384 03386

:= 51 × (h3! k 3!) − 0!i := 3!! + 43 × h3! k 0!i := 55 × h3! k (3 × 0)!i := 4! × h(8 + 3!) k (3 × 0)!i := 6! + h8 k 3!i × h3 k 0!i

D√ E 03953 := −3! + 5! × 9 k 3 − 0! √ √ 4+5 !× 9 !! − h3! k 0!i 03954 := D√ E 03961 := (−1 + 6)! × 9 k 3 + 0! D √ E3−0! 03963 := −3! + 6 k 9 03964 := 4 × (6! + h(9 × 3) k 0!i) 03965 := (56 + 9) × h3! k 0!i √ 9 !! + h3! k 0!i 03966 := −6! + 6 × √ 03968 := 8 + 6!/ 9 ! × h3 k 0!i D √ E 03984 := 4! × 8 k 9 × (3 − 0!) D E √ 03999 := 9 + 9 k 9 × h3 k 0!i 22

Inder J. Taneja


√ 04059 := 9 × 5! + 0! × h4 k 0!i √ 04069 := 9 ! × 6! − h(0! + 4!) k 0!i √ 9 ! × (7 − 0!)! − h4! k 0!i 04079 := √ 04159 := − 9 !! + (5! − 1) × h4 k 0!i

04623 := (h3! k 2i + 6) 4 − 0! √ 04674 := − 4 + 7 ! − 6 × h4 k 0!i √ D √ E 04693 := 3! × 9 !! + 6 k 4 + 0!

04163 := h(3 × 6) k 1i × (4! − 0!)

04709 := −90 + 7! − h4! k 0!i

04179 := −

√ 9 !! + 7! − h14 k 0!i

04236 := h6 k 3!i2 − (4 + 0!)! √ 04239 := 9 ! × 3!! − h(2 × 4) k 0!i √ √ D√ E 04299 := 9 !× 9×2 !− 4 k 0! 04333 := 33! × 3! − h4 k 0!i 04337 := (7 − 3)3! + h4! k 0!i 04338 := (8 × 3 − 3!) × h4! k 0!i 04353 := 3 × (5 + 3! × h4! k 0!i) 7

04373 := 3! / h3! k 4i − 0! 04386 := 6 × (8 + 3 × h4! k 0!i) √ D√ E 04389 := 9 ! × (8 + 3!!) + 4 k 0! D E √ 04393 := 3! × 9 k 3 × 4! + 0! 04422 := 22 × h(4! − 4) k 0!i 04437 := 7! − 3 × h(4! − 4) k 0!i

04599 := (99 + 5!) ×

D√

4 k 0!

E

√

04718 04727 04733 04736 04754

:= := := := :=

−81 + 7! − h4! k 0!i −72 + 7! − h4! k 0!i (− h3! k 3!i + 7! − h4! k 0!i) −63 + 7! − h4! k 0!i −45 + 7! − h4! k 0!i

04757 := −7!/5! + 7! − h4! k 0!i 04763 := −36 + 7! − h4! k 0!i D√ E 04767 := 7! − (6 + 7) × 4 k 0! 04772 := −27 + 7! − h4! k 0!i 04781 := −18 + 7! − h4! k 0!i 04807 := 7! + 08 − h4! k 0!i 04824 := 4! × h(2 × 8 + 4) k 0!i 04827 := 7! + 28 − h4! k 0!i D √ E 04836 := 6 × 3!! + 8 k 4 + 0! ! 04837 := 7! + 38 − h4! k 0!i

04439 := 9!/34 − h4 k 0!i 04455 := 55 × h(4 + 4) k 0!i 04471 := 17 × (h4! k 4!i − 0!) D√ E 4 k 0! 04473 := (3 + 7!/4!) × D E D√ E √ 04494 := 4! − 9 k 4 × 4 k 0! √ 04579 := 9 + 7 ! − 5 × h4! k 0!i 04584 := (−4 + 8)! × h(−5 + 4!) k 0!i 04586 := −6 + (−8 + 5!) × h4 k 0!i √ 04589 := − 9 + (−8 + 5!) × h4 k 0!i √ 9 04592 := −2 + 5! × h4 k 0!i √ 04596 := 6 × 9 !! + 5 + h4 k 0!i

04838 := h(8 + 3) k 8i × h4 k 0!i 04867 := 7! + 68 − h4! k 0!i D√ E 04871 := −1 + 7! − 8 × 4 k 0! D E √ 04873 := −3! + 7! − 8 × 4 k 0! D√ E √ 04874 := 4 + 7! − 8 × 4 k 0! 04877 := 7! + 78 − h4! k 0!i √ D E √ 04879 := 9 !! × 7 − 8 × 4 k 0! D E √ 04887 := 7! + 8 − 8 × 4 k 0! √ 04889 := 9 !! + 8! /8 − h4! k 0!i √ 9 ! + 0! ! − h(9 + 4) k 0!i 04909 := 23

Inder J. Taneja


√ 04917 := 7! − 1 × 9 × h4 k 0!i √ D√ E 04919 := 9 !+1 !− 9 × 4 k 0!

05466 := 6 × (6! + h(4! − 5) k 0!i) D√ E 05467 := 7 × 6! + 4 + 5 ! k 0!

04927 := 7! − h(2 + 9) k 4) + 0! E D √ 04929 := (9 − 2)! − 9 + 4 k 0! E √ D√ 04935 := −5 + 3!!/ 9 × 4 k 0!

05522 := 22 × h(5 × 5) k 0!i 05544 := 4! × h(4! − 5/5) k 0!i q√ 98 × (5! − h5 k 0!i) 05589 :=

D √ E 04937 := 7! − 3! k 9 − 40 √ 04961 := 1 + 6!/ 9 ! × h4 k 0!i D√ √ E 04972 := −2 + 7! − 9 !k 4 + 0! ! D √ √ E 04987 := 7! − 8 − 9 k 4 − 0! D E √ 04995 := 5 × 9 × 9 + 4 k 0! √ 04999 := 9 ! + 9/9 ! − h4 k 0!i D√ E 05009 := − 9 k 0! + (0! + 5 + 0!)! √ D√ E 05019 := 9 !+1 !− −0! + 5 k 0! 05027 := 7! − 2 − h0! k (5 × 0)!i 05029 := (9 − 2)! − h0! k (5 × 0)!i 05041 := (1 + h4 k 0!i) × 5! + 0! E D√ 4 k 0! × 5! + 0! 05042 := 2 × D√ E 05061 := (1 + 6)! + −0! + 5 k 0! 05081 := (−1 + 8)! + h(−0! + 5) k 0!i 05091 := (−1 + 9 − 0!)! + h5 k 0!i 05092 := (−2 + 9)! + 0! + h5 k 0!i D√ E 05147 := 7! − 4 + 1 + 5! k 0! √ 05191 := 1 + 9 ! ! + h15 k 0!i √ 9 ! + h15 k 0!i 05197 := 7! + D √ 05248 := 8 k 4 × 25+0! √ 05291 := 1 + 9 ! ! + h25 k 0!i √ 05297 := 7! + 9 ! + h25 k 0!i √

05329 := h9 − 2 k 3i 5−0! DD E √ E 05424 := 4! × 2 k 4 k 5 + 0!

05709 := (9 − 0!)!/7 − h5 k 0!i 05784 := 4! × h(8/ (7 − 5))! k 0!i D √ E 05793 := 3 k 9 + 7! + (5 + 0!)! 05803 := (3!! − 0!) × 8 + h5 k 0!i √ 05819 := 9 !! + 1 × 8 + h5 k 0!i 05843 := (3!! + 4) × 8 + h5 k 0!i 05867 := (7 + 6!) × 8 + h5 k 0!i 05895 := 5 × 9 × h(8 + 5) k 0!i D √ E 05904 := 4! k 09 ! × (5 − 0!)! √ 05964 := 4 × 6! + 9 !! + h5 k 0!i 05967 := (7 + 6) × 9 × h5 k 0!i 06239 := 9 × 3!! − h(−2 + 6)! k 0!i √ 06344 := h4! k 4i × 36 − 0! 06393 := 3!! + 93 × h6 k 0!i 06419 := 9 × (−1 + 4)!! − h6 k 0!i 06439 := 9 × 3!! − h4 k (6 × 0)!i 06495 := −5 + 94 − h6 k 0!i D √ E √4 06497 := 7 k 9 ! + 6! + 0! 06539 := 9 × 3!! + 5! − h6 k 0!i 06655 := 55 × h(6 + 6) k 0!i 06744 := 4! × h(4 × 7) k (6 × 0)!i 06832 := ((2 + 3)! − 8) × h6 k 0!i √ 06952 := −2 + 5! − 9 ! × h6 k 0!i √ 06955 := −5 + 5! × − 9 + h6 k 0!i D √ E 06993 := 3! k 9 × (−9 + (6 − 0!)!) 24

Inder J. Taneja


√ 07299 := 9 × 9 !! + h(2 + 7) k 0!i √ 9 ! + 5! × h7 k 0!i 07597 := −7 −

09834 := − h4! k 3!i + 8!/

√

9 + 0!

07755 := 55 × h(7 + 7) k 0!i 07839 := 9 × (3!! + h(8 + 7) k 0!i) 07865 := (−5 + 6!) × h(8 − 7) k 0!i

09919 := h(9 + 1) k 9i × h9 k 0!i 09955 := 55 × h(9 + 9) k 0!i E D√ 9 ! + 9 k 0! 09966 := 66 × E D√ 9 ! k 0! 09984 := 4! × 8 × −9 +

√ 07952 := −2 + 5! − 9 ! × h7 k 0!i √ √ 07995 := − 5! + 9 × 9 ! − h7 k 0!i

√ √ 09989 := 9 !! × 8 + 9 ! − h9 k 0!i 10275 := 5 × 7 + 2h0!k1i

08019 := 9 × h1 k 0!i × h8 k 0!i 08343 := h3! + 4 k 3i × h8 k 0!i 08424 := (4! + 2) × 4 × h8 k 0!i

10582 := (2 + 8 × 5!) × h0! k 1i 10584 := 4! + 8 × 5! × h0! k 1i √ 10593 := 3!! + 95 × h0! k 1i

D E √ 08439 := (9! − 3) / 4 k 8 + 0!

10635 := 5 × 3 × (6! − h0! k 1i) √ 10789 := 9 !! × 8 + (7! − h0! k 1i) √ 10919 := 9 ! − 1 ! × h9 k 0!i − 1 √ 10997 := 7 + 9!/ 9 / h0! k 1i √ √ 10998 := 8! + 9 ! × 9/ h0! k 1i

08444 := 4 × (h4! k 4!i × 8 − 0!) D √ E 08694 := 4! k 9 ! + 6! × (8 + 0!) q 08844 := 44 × 8! + h8 k 0!i 08855 := 55 × h(8 + 8) k 0!i √ 08964 := 4 × 6! × 9 + h8 k 0!i √ 08974 := 4 × (7 + 9!/ h8 k 0!i) √ 08991 := −1 + 9 ! ! − 9 × h8 k 0!i D √ E 09159 := (9 + 5!) × 1 + 9 ! k 0! 09168 := 8 × 6 × h19 k 0!i 09282 := h(2 + 8) k 2i × h9 k 0!i D √ √ E 09348 := − 8 k 4 × 3! − 9 ! − 0! ! 09373 := h(3 + 7) k 3i × h9 k 0!i 09456 := 6! + (5! − 4!) × h9 k 0!i 09464 := h(4 + 6) k 4i × h9 k 0!i 09555 := h(5 + 5) k 5i × h9 k 0!i 09646 := h(6 + 4) k 6i × h9 k 0!i √ 09664 := 46 × h(6 + 9) k 0!i 09737 := h(7 + 3) k 7i × h9 k 0!i 09828 := h(8 + 2) k 8i × h9 k 0!i

:= (5 × h2 k 0!i)1+1 := 8 × 6 × h(5 − 1)! k 1i := 4!!/ h(4 × 1 − 2) k 1i! := 4! × h8 k 3!i × (2 + 1)! √ 13176 := 67−1 × h3! k 1i

11025 11568 12144 12384

13255 := 55 × h(−2 + 3!)! k 1i √ D √ E 4 13448 := 8 k 4 × (3 − 1) D√ E 9 ! k 1 − 2 × h4! k 1i 14219 := √ 14379 := 9 × (7! − 3! − h4! k 1i) √ 14397 := 7! × 9 − 3 × h4! k 1i √ 14455 := −5 + 5!/ 4 × h4! k 1i √ D√ E 14616 := 6! − 16 ! × 4k1 14647 := 74 × 6 + h4! k 1i √ D√ E 14679 := − 9 × 7 + 6! × 4k1 D √ E 14873 := 3 × 7! − 8 k 4 − 1 25

Inder J. Taneja

14879 := 14931 := 14942 := 14992 := 14994 :=


√

9 × 7! − h(8 − 4)! k 1i D√ E 4k1 (1 × 3!! − 9) × √ −2 + 4 9 × h4! k 1i √ √ D√ E −2 + 9 !! − 9 ! × 4k1 √ D√ E √ 4×9 !− 9 ! × 4k1

D√ E 19375 := 57−3 × 9k1 E D√ 9 9k1 19592 := 2 + 5! × D √ E D √ E 19844 := 4! k 4 × 8 k 9 − 1 √ 9 9 −1 19924 := h4! k 2i + √ D√ E 19974 := 4 × 7! − 9 !× 9k1

√ 15079 := 9 × 7! − h(−0! + 5) k 1i 15096 := (6! − h9 k 0!i) × (5 − 1)! E D√ 4 k 1 × (5 + 1) ! 15144 := 4! + D√ E 15162 := (2 + 6!) × −1 + 5 k 1 E D√ 25 k 1 15273 := 3 × 7! +

√ 20048 := 8!/ 4 − hh0! k 0!i k 2i 20147 := 7! × 4 − h1 k 0!i − 2 √ D√ E 20909 := 9 !! + 0! × 9 k 0! − 2 E D√ 9 k 1 × 3!! − 2/2 22319 := 2 22338 := 8! + h3! k 3!i /2

15436 := 63 × h4! k 5i + 1 15477 := 77 × h(4 × 5) k 1i D √ E 15696 := − 6 k 9 ! + 6! × (5 − 1)!

23266 := 66 /2 − h3! k 2i

15851 := (1 + 5!) × (h8 + 5 k 1)) 16104 := 4! × h0! k 1i × 61

D √ E 16446 := 6 k 4 + 46+1 D √ E 16742 := 2 k 4 × 761 16743 := − h3! k 4i + 76+1 √ D√ E 17249 := 9 !! × 4! − 2+7 k 1

23409 := h(−9 + (04)!) k 3i2 D √ E 9 ! × (−4 + 3!!/2) 23496 := 6 k D E √ 23593 := 3 k 9 × (−5 + 3!!) − 2 23874 := −47 + 8! − h3! k 2i .

√ E 23994 := 4 k 9 × 9 × h3! k 2i D

24334 := 43! × 3! − h4! k 2i D 2 √ E 24649 := 9 k 4 − 6! /4 24964 := (4 + h(6 + 9) k 4i)2

17324 := h4! k (−2 + 3!)i × 71

26896 := (h(6 + 9) k 8i + 6)2

17346 := 6! × 4! + h3! k (7 − 1)i D √ E2 18244 := 4! k 4 − 8! × 1 D√ E 18264 := 4! × 6! + 2×8 k 1

27883 := −3! + h(8 + 8) k 7i2 D√ E2 27889 := 9×8−8 k 7 D √ E 28644 := 4 k 4 × 682

18544 := h4! k 4i × (−5 + 81) D√ E 19344 := (−4! × 4 + 3!!) × 9k1

28764 := (− h4! k 6i + 7!) × (8 − 2) D √ E 29405 := (5! + 0!) × 4! k 9 + 2 26

Inder J. Taneja


29524 := h4! k 2i ×

5! +

r

√ 9 !−2

!

D E √ √ 29748 := − 8 k 4 + 7! × 9 × 2 D√ E √9 −2 29789 := 9 k (8 − 7) D √ E √9 29793 := 3 k − 9 !+7 +2 √ 29808 := h8 k 0!i × 8 + 9 !!/2

33768 := 8 × 67 × h3! k 3i √ 33924 := 4 + 29 × h3! k 3!i D √ E 34398 := 8! + 9 × 3! k 4 − 3!! 34484 := −4 + 8! − 4! × h4! k 3i D √ E 34486 := 6! + 8 k 4 × 43

√ E √4 34593 := 3! k 9 ! + 5! −3 √4 D√ √ E 9 !k 9 ! + 5! +3 34599 := D

√ √ E 9 !! /2 29848 := 8 k 4 × 8 + D √ E √ 9 ! + 9 ! × (9 − 2) ! 29994 := − 4! k D

30172 := −2 + (7! − h1 k 0!i) × 3! √ 30174 := 4 × (7! − h1 k 0!i) × 3 D√ E 30282 := 2 × 8 k 2 × (0! + 3!!)

34704 := h4! k (0 × 7)!i × 4! × 3! 35448 := 8! − 4! × h(4 × 5) k 3i 35777 := 7 × (7! + h7 k (−5 + 3!)i)

35939 := 30378 := (−8 + 7! + h3 k 0!i) × 3! 30473 := 3! × 7! + h(4! − 0!) k 3i √ 30474 := − 4 + 7! + h4 k 0!i × 3! 30476 := 6 × 7! + h(4! − 0!) k 3!i 30478 := −8 + (7! + h4 k 0!i) × 3!

30504 := h4 k 0!i × ((5 − 0!)! + 3!!) 30606 := ((6 + 0!)! + h6 k 0!i) × 3! 30636 := (h6 k 3!i + (6 + 0!)!) × 3! √ 30879 := 9 ! × 7! − h8 k 0!i + 3!! D√ E3 √ 30897 := −7! + 9 k 8 + 0!

D √ E 31899 := 9 k 9 × (8 − 1)3 32085 := 5 × h8 k 0!i2 − 3!! D E2 √ 32448 := 8+ 4 k 4 ×3 32452 := (2 + 5!) × h(4! + 2) k 3!i 33534 := h4! k 3i × (5! + 3 × 3!)

D√

9k3

E √9

+5−3

36432 := 23 × 4! × h6 k 3!i D√ E3 37044 := 4 × 4 k (0 × 7) ! D √ E2 37297 := 7 k 9 × 7 − 3! D √ E 37453 := 3!! × 5 k 4 + 7 + 3!

37536 := 63 ! − 5! × h7 k 3!i E √ D √ 37596 := 6! + 9 × 5 k 7 − 3 D E √ 38155 := −5 + 5 k 1 + 8 × 3!! D √ E 38428 := 8! − 2 k 4 × h8 k 3!i 38634 := − h4! k 3!i − 6! + 8! − 3!!

38966 := 6! × 6 × 9 + h8 k 3!i 39105 := 5 × h0! k 1i × (−9 + 3!!) D √ E3 39302 := −2 + 0! + 3 k 9 39304 := (h4 k 03i − 9)3 D√ E 39375 := 57−3 × 9 !k3 27

Inder J. Taneja


D√ E 39468 := 8! − 6! − 4 × 9k3 D √ E 39498 := 8! − 9 k 4 × 9 + 3! √ 39509 := − h9 k 0!i + 5 + 9 ! − 3!! D √ E 39658 := 8! − 5 + 6 k 9 − 3!! D √ E 39687 := 7!/8 × 6 k 9 − 3

39688 := −8 + 8! − 6! + h9 k 3!i D√ E 39786 := −6 + 8 × 7! − 9 ! k 3! D√ E √ 9 ! k 3! 39789 := − 9 + 8 × 7! − D√ E √ 39792 := 2 9 × 7! − 9 ! k 3! E D √ 39828 := 8! − 2 × 8 × 9 k 3!

√

40199 := 9!/9 − h1 k 0!i 4 D √ 40208 := 8! − h0! k (2 × 0)!i k 4 D √ E 40238 := 8! − (3! + 2) k 04 D √ E 40258 := 8! − (5 − 2)! k 04 D √ E 40268 := 8! − (6 − (2 × 0)!) k 4

√ 40353 := (3 + 5)! + h3 k 0!i + 4 D √ E 40362 := (2 + 6)! + (3 + 0!) k 4 √ 4 + 6 ! + h(3 + 0!) k 4i 40364 := 40378 := 8! − 7 + h3! k 0!i + 4 40381 := 1 × 8! + h3! k (0 × 4)!i

√

:= h4! k 3i + 8! − 93 := h4! k 4i + 8! − (9 − 3)! := 8! + h4! k 8i − (9 − 3)! := h4! k 5i + 8! + 9 − 3!! D√ E √ 9 + 7!/8 × 9 !k3 39879 := 39834 39844 39848 39854

D√ √ E 39918 := 8! − 1 + 9 !k 9 ! × 3! D√ √ E 39924 := (4 × 2)! − 9 !k 9 ! × 3! D√ √ E 39942 := (2 × 4)! − 9 ! k 9 × 3! 39948 := 8! − 4 × h9 k (9/3)i 40058 := 8! − (5! + h0! k 0!i) ×

√ 4

D √ E 40078 := 8! − (−7 + h0! k 0!i)! k 4 D √ E 40108 := 8! − h(0! + 1) k 0!i k 4 D √ E 40158 := 8! − 5 + h1 k 0!i k 4 D √ E 40184 := −4! + 8! − h1 k 0!i k 4 40185 := −5! + 8! + h1 k 0!i + 4

√ E 40384 := 4 + 8! + 3! k 04 √ 40394 := 4!/ 9 ! + h(3! + 0!) k 4i D E √4 √ 40401 := 10 × 4 k 0! D

40479 := −9 + 7 × h4! k 0!i × 4! 40524 := (4 × 2)! + h5 k 0!i × 4

40548 := 8! + 4! + h5 k 0!i × 4 √ 40564 := 4 + 6 ! + h(5 − 0!)! k 4i √ √ 9 ! × h6 k 0!i + 4 40698 := 8! + √ 40869 := − 9 + 6! × (h8 k 0!i − 4!) D√ E 40926 := (6! − 2) × 9 ! k 0! − 4

D √ E 40938 := 8! + 3!! − (9 + 0!) k 4 40978 := 8! + 7 × h9 k 04i D√ E 40983 := 3!! + 8! − 9 ! k 0! + 4 √ D√ √ E 40998 := 8! + 9 !! − 9 + 0! k 4 42384 := 4! × h8 k 3!i + (2 × 4)! 28


Inder J. Taneja

43533 := h3! k 3i × (−5 + 3!! − 4!) D √ E 43648 := (8 − 4! + 6!) × 3! k 4 43856 := 6! × 5 + 8! − h3! k 4i √ 9 !! − 3 − 4 43913 := h3! k 1i × D√ E √ 43919 := 9 !k1 × 9 !! + 3 − 4

44164 := h(4! − 6) k 1i × h4! k 4i √ D √ E √ 44394 := −4 + 9 !! × 3! k 4 + 4 √ D √ E 44398 := 8! + 9 ! × 3!! − 4! k 4 √ 44469 := 96 × h4! k 4i /4 D √ E 44528 := 82 + 5! × 4! k 4

√ E 44616 := 6! × (1 × 6) k 4 − 4! D √ E 44633 := −3 + 3!! × 6 k 4 − 4 D √ E √ 44634 := −4 + 3!! × 6 k 4 − 4

46344 := h4! k 4!i + 3!! × 64 46384 := − h4! k 8i + 3!6 − 4! 46535 := −5 + h3! k 5i × (6! − 4) q √ 46605 := − h5 k 0!i + 6(6× 4) D√ E 46619 := − 9 ! k 1 + 66 + 4!

D √ E 46693 := 3 k 9 + 66 + 4 √ 46944 := h4! k 4!i + 9 !6 + 4! D√ E 46968 := (−8 + 6!) × 9 ! k 6 − 4! √ D √ E 46978 := 8! × 7/ 9 !− 6 k 4 E √4 D√ 9 k (8 × 0) ! × 7 47089 :=

D

44636 := 6! × h3! k (6 − 4)i − 4 D √ E √ 44639 := 9 + 3!! × 6 k 4 − 4 D √ E √ 44642 := (2 + 4)! × 6 k 4 + 4 D √ E 44644 := (4!/4)! × 6 k 4 + 4 D √ E √ 44646 := 6 k 4 × 6! + 4 + 4 D √ E 44662 := −2 + 6! × 6 k 4 + 4! √ D √ E 44764 := 4 + 6! × (7 − 4)! k 4 D √ E 44918 := 8! + 19 × 4! k 4 √ D √ √ E 4 44942 := −2 + 4! − 9 k 4 D √ E 45308 := 8! + (0! + 3!)! − 5 k 4 √ 45696 := 6! − 9 ! × h(6!/5!) k 4i D√ √ E 9 ! k 9 × (6! + 5) + 4! 45699 :=

√ 47334 := (h4! k 3!i + 3!!) × 74 47424 := 4 × 2 × h4! k 7i × 4! 47448 := 8 × 4! × h4! k 7i + 4! √

47524 := (h(4! − 2) k 5i − 7)

√

49005 := 5 × (h0! k 0!i × 9)

4

4

√ 49236 := h6 k 3!i × 2 + 9 !! + 4! D√ √ E 49675 := −5 + 7! + 6! × 9 !k 4 D √ E √4 49923 := 3 × 2 × 9 ! k9 50765 := (−5 + 6!) × h7 k (0 × 5)!i √ 50907 := h7 k 0!i × − 9 + (0! + 5)!

51479 := 9!/7 − h4! k 1i − 5! 53694 := (h4! k 9i − 6!) × (3! − 5!) 54336 := 63 ! + h3! k 4i × 5! D√ E 57148 := 8! + 4 k 1 + 75 √ D √ E −4+8 58444 := 4! k 4 − 5! 29

Inder J. Taneja


59014 := −4! − h1 k 0!i + 95 D√ E 59018 := − 8 + 1 k 0! + 95 59024 := −4 − h2 k 0!i + 95 E D√ 8/2 k 0! + 95 59028 := − D√ E 59109 := 9 ! k 0! − 1 + 95

69549 :=

D

9k

√ E √ 4 + 5 × − 9 + 6!

70682 := 2 × (8! + h6 k 0!i − 7!) 71065 := 56 + h0! k 1i × 7! D √ √ E 9 !! × 9 k 9 − 1 + 7! 71999 := 73444 := h4! k 4i × 43 × 7

59134 := 4! + h3! k 1i + 95 D √ E 59291 := 1 + 9 ! k 2 + 95 D √ E 59294 := 4! k 9 + 2 + 95

74347 := 7 × 43 × h4! k 7i 77448 := 8! + (h4! k 4!i + 7!) × 7 D √ E 79989 := 9! − 8! + 9 k 9 × 7 D√ E 80472 := 2 × 7! − 4 k 0! × 8

59345 := h(5 + 4!) k 3!i + 95

80529 := −9 + 2 × (− h5 k 0!i + 8!)

59433 := 3! × h3! k 4i + 95

E D√ 9 ! k 4 + 95 D√ E 9 59938 := 8! + 3 − 9 !k5 59497 := 7 ×

60439 := 9!/3! − h4 k (0 × 6)!i 62436 := (6! + 3!) × h4 × 2 k 6i D √ E 2 62494 := 4! k 9 ! +4 −6

62638 := h8 k 3!i × 6! − 2 + 6! D E √ 63888 := 8 k 8 × 8 × (3! + 6!) D√ E 64728 := 82 k 7 × (4! + 6!) √ 64944 := 4! × 4 + 9 × h4! k 6i D √ E 64984 := 48 − 9 k 4 × 6

D √ E 66898 := 8 k 9 × (86 + 6!) D √ E 66954 := 4 + 5 k 9 × 6! − 6 D √ E 66996 := 6! × 9 k 9 + 6 × 6 68395 := −5 + h9 k (−3 + 8)i × 6! √ 69399 := −9 + h9 k 3!i × 9 + 6!

80532 := 2 × (−3 − h5 k 0!i + 8!) 80538 := (−8 + 3!) × (h5 k 0!i − 8!) 80583 := −3! + 8! − h5 k 0!i + 8! √ 80589 := 9 × 8! − h5 k 0!i − 8! 80598 := 8! + 9 − h5 k 0!i + 8!

80662 := 2 × (h(6/6) k 0!i + 8!) 80682 := 2 × (h(8 − 6) k 0!i + 8!) √ 80698 := 8! − 9 + h6 k 0!i + 8! 80728 := 8! × 2 + h(7 + 0!) k 8i D √ E 80742 := 2 × − 4 + 7 k 0! + 8!

80762 := 2 × (h6 k (7 × 0)!i + 8!) 80782 := 2 × (8! + h7 k (0 × 8)!i) √ √ 80794 := 4 × 9 ! + h7 k 0!i + 8! √ 80804 := 4 × (0! + h8 k 0!i + 8!) D√ E 80842 := 2 × 4 + 8 k 0! + 8!

80883 := 3 × (8! + h8 k 0!i) − 8! 80942 := 2 × (h(4! − 9) k 0!i + 8!) 80982 := 2 × (h(8 + 9) k 0!i + 8!) D √ E 85344 := 4 k 4 + 3!! × (5! − 8) 30


Inder J. Taneja

87744 := (4! × h4! k 7i + 7!) × 8 √ 90567 := (7! × 6 − h5 k 0!i) × 9 92364 := (−4 + 6!) × h(3! × 2) k 9i √ 92493 := 3!! − 9 × h(4!/2) k 9i √ 93024 := − 4 + h2 k 0!i !/ (3! + 9)! √ 3! 93504 := − h4 k 0!i + 5 × 9 !

94058 94794 96624 98404 99645

3

√ D √ E 9 !! + 3! k 9 √ √ 9 !! := h(8 + 5) k 0!i × − 4 + D √ E √ := 4 × 9 !/7! − 4! k 9 ! D √ E 9 ! := (4! + 2 × 6!) × 6 k √ √ := h4! k 0!i 4 + 8! + 9 √ := 5 × h4! k 6i + 99

93955 := −5 + 5! ×

Number Patterns

In 1966, Madachy [14] pp. 174-175, gave an idea of number patterns writing as: 34 × 425 := 34425 4

3 × 4250 := 344250 4

3 × 42500 := 3442500

312 × 325 := 312325 2

31 × 3250 := 3123250 312 × 32500 := 31232500

73 × 9 × 42 := 73942 73 × 9 × 420 := 739420 73 × 9 × 4200 := 7394200

Based on same idea as above, there are some concatenation-type selfie numbers, those can be extended in patterned form just multiplying successively by 10. See below examples: E D √ 305 := h3! k 0!i × 5 1968 := 1+ 9 ! k 6 ×8 D E √ 3050 := h3! k 0!i × 50 19680 := 1 + 9 ! k 6 × 80 30500 := h3! k 0!i × 500 D E √ 196800 := 1 + 9 ! k 6 × 800 D √ E 396 := 3! k 9 ! ×6 D √ E 3960 := 3! k 9 ! × 60 D√ E D √ E 3905 : = 3!! + 9 ! k 0! ×5 39600 := 3! k 9 ! × 600 E D√ D √ E 39050 := 3!! + 9 ! k 0! × 50 D√ E 492 := 4! k 9 ! ×2 D √ E 390500 := 3!! + 9 ! k 0! × 500 4920 := 4! k 9 ! × 20 D √ E 49200 := 4! k 9 ! × 200 10285 := h1 k 0!i2 × 85

√ E 1446 := −1 + 4! k 4 × 6 D √ E 14460 := −1 + 4! k 4 × 60 D √ E 144600 := −1 + 4! k 4 × 600

D

102850 := h1 k 0!i2 × 850 1028500 := h1 k 0!i2 × 8500

1476 := (−1 + h4! k 7i) × 6 14760 := (−1 + h4! k 7i) × 60 147600 := (−1 + h4! k 7i) × 600

10635 := (− h1 k 0!i + 6!) × 3 × 5 106350 := (− h1 k 0!i + 6!) × 3 × 50 1063500 := (− h1 k 0!i + 6!) × 3 × 500 31


Inder J. Taneja

10648 := h1 k 0!i

√ 6/ 4

106480 := h1 k 0!i6/ 1064800 := h1 k 0!i6/

√ √

30667 := (h3! k 0!i + 6! × 6) × 7 306670 := (h3! k 0!i + 6! × 6) × 70 3066700 := (h3! k 0!i + 6! × 6) × 700

×8

4

× 80

4

× 800 33044 := (3!! + h3 k 0!i) × 44 330440 := (3!! + h3 k 0!i) × 440 3304400 := (3!! + h3 k 0!i) × 4400

14056 := h(1 + 4!) k 0!i × 56 140560 := h(1 + 4!) k 0!i × 560 1405600 := h(1 + 4!) k 0!i × 5600

33124 := h(3 × 3) k 1i2 × 4 15129 := h(−1 + 5) k 1i2 × 9

331240 := h(3 × 3) k 1i2 × 40

151290 := h(−1 + 5) k 1i2 × 90

3312400 := h(3 × 3) k 1i2 × 400

1512900 := h(−1 + 5) k 1i2 × 900 33327 := (h3! k 3i + 3!)2 × 7 19225 :=

D√ 9 !k2 1+

E2

333270 := (h3! k 3i + 3!)2 × 70

×5

3332700 := (h3! k 3i + 3!)2 × 700

E2 D√ 9 !k2 × 50 192250 := 1 + D√ E2 1922500 := 1 + 9 !k2 × 500

39105 := (3!! − 9) × h1 k 0!i × 5 391050 := (3!! − 9) × h1 k 0!i × 50 3910500 := (3!! − 9) × h1 k 0!i × 500

25575 := (h(2 + 5) k 5i + 7!) × 5 255750 := (h(2 + 5) k 5i + 7!) × 50 2557500 := (h(2 + 5) k 5i + 7!) × 500

√ E 39768 := − 3! k 9 + 7! − 6 × 8 D √ E 397680 := − 3! k 9 + 7! − 6 × 80 D √ E 3976800 := − 3! k 9 + 7! − 6 × 800

30576 := (h3! k 0!i − 5 + 7!) × 6 305760 := (h3! k 0!i − 5 + 7!) × 60 3057600 := (h3! k 0!i − 5 + 7!) × 600

30636 := ((3! + 0!)! + h6 k 3!i) × 6 306360 := ((3! + 0!)! + h6 k 3!i) × 60 3063600 := ((3! + 0!)! + h6 k 3!i) × 600

E

39052 := 3!! + 9 k 0! × 52 D√ E 390520 := 3!! + 9 k 0! × 520 E D√ 9 k 0! × 5200 3905200 := 3!! +

25405 := ((2 + 5)! + h4 k 0!i) × 5 254050 := ((2 + 5)! + h4 k 0!i) × 50 2540500 := ((2 + 5)! + h4 k 0!i) × 500

30606 := ((3! + 0!)! + h6 k 0!i) × 6 306060 := ((3! + 0!)! + h6 k 0!i) × 60 3060600 := ((3! + 0!)! + h6 k 0!i) × 600

D√

D

40959 := h4 k 0!i × (−9 + 5!) × 9 409590 := h4 k 0!i × (−9 + 5!) × 90 4095900 := h4 k 0!i × (−9 + 5!) × 900 √

D√ E 45909 := 4+5 !+ 9 ! k 0! × 9 √ D√ E 459090 := 4+5 !+ 9 ! k 0! × 90 √ D√ E 4590900 := 4+5 !+ 9 ! k 0! × 900 32

Inder J. Taneja

RGMIA Research Report Collection, 20(2017), pp. 1-42, http://rgmia.org/v20.php √ E 9 + 4!3 × 5 D √ E 3 694350 := 6 k 9 + 4! × 50 D √ E 6943500 := 6 k 9 + 4!3 × 500

47424 := h4! k 7i × 4! × 2 × 4 474240 := h4! k 7i × 4! × 2 × 40 4742400 := h4! k 7i × 4! × 2 × 400

69435 :=

47488 := (h4! k 7i × 4! + 8) × 8 474880 := (h4! k 7i × 4! + 8) × 80 4748800 := (h4! k 7i × 4! + 8) × 800

D

6k

√

80482 := − h8 k 0!i + 4 + 8! × 2 √ 804820 := − h8 k 0!i + 4 + 8! × 20 √ 8048200 := − h8 k 0!i + 4 + 8! × 200

D E2 √ 49923 := (4 × 9) k 9 × 3 D E2 √ 499230 := (4 × 9) k 9 × 30 E2 D √ 4992300 := (4 × 9) k 9 × 300

80802 := (h8 k 0!i + 8!) × 02 80802 := (h8 k 0!i + 8!) × 020 80802 := (h8 k 0!i + 8!) × 0200

60846 := (h6 k 0!i + 8!/4) × 6 608460 := (h6 k 0!i + 8!/4) × 60 6084600 := (h6 k 0!i + 8!/4) × 600

80832 := (8! + h(0! + 8) k 3!i) × 2 808320 := (8! + h(0! + 8) k 3!i) × 20 8083200 := (8! + h(0! + 8) k 3!i) × 200

63084 := (6! + h3 k 0!i) × 84 630840 := (6! + h3 k 0!i) × 840 6308400 := (6! + h3 k 0!i) × 8400

D√

E 9 k 0! + 7! + 8! × 2 D√ E 907820 := 9 k 0! + 7! + 8! × 20 E D√ 9 k 0! + 7! + 8! × 200 9078200 := 90782 :=

68085 := (6! + h8 k 0!i) × 85 680850 := (6! + h8 k 0!i) × 850 6808500 := (6! + h8 k 0!i) × 8500

E D√ 9 k 0! × 92 69092 := 6! + D√ E 690920 := 6! + 9 k 0! × 920 D√ E 6909200 := 6! + 9 k 0! × 9200

91295 := 912950 := 9129500 :=

√ 69408 := 6 × 9 ! × h4! k 0!i × 8 √ 694080 := 6 × 9 ! × h4! k 0!i × 80 √ 6940800 := 6 × 9 ! × h4! k 0!i × 800

D√ D√ D√

9k1 9k1 9k1

E2 E2 E2

× 95 × 950 × 9500

D√ √ E2 9 ! k 9 × 25 D√ √ E2 9 ! k 9 × 250 992250 := D√ √ E2 9922500 := 9 ! k 9 × 2500 99225 :=

33

Inder J. Taneja

4


Summary: Selfie Numbers

This section, we shall give some idea of selfie numbers calcuated in different situations. These are divided in subsection as below.

4.1

Factorial

This subsection brings selfie numbers with use of factorial. See below some examples: 145 = 1! + 4! + 5! 733 = 7 + 3!! + 3! 5177 = 5! + 17 + 7!.

363239 = 36 + 323 + 9! 363269 = 363 + 26 + 9! 403199 = 40319 + 9!.

1463 = −1! + 4! + 6! + 3!!. 10077 = −1! − 0! − 0! + 7! + 7!. 40585 = 4! + 0! + 5! + 8! + 5!. 80518 = 8! − 0! − 5! − 1! + 8!. 317489 = −3! − 1! − 7! − 4! − 8! + 9!. 352797 = −3! + 5 − 2! − 7! + 9! − 7!. 357592 = −3! − 5! − 7! − 5! + 9! − 2!. 357941 = 3! + 5! − 7! + 9! − 4! − 1!.

361469 = 3! − 6! − 1! + 4! − 6! + 9!. 364292 = 3!! + 6! − 4! − 2! + 9! − 2!. 397584 = −3!! + 9! − 7! + 5! + 8! + 4!. 398173 = 3! + 9! + 8! + 1! − 7! + 3!. 408937 = −4! + 0! + 8! + 9! + 3!! + 7!. 715799 = −7! − 1! + 5! − 7! + 9! + 9!. 720599 = −7! − 2! + 0! − 5! + 9! + 9!.

For details refer author’s work [30, 31].

4.2

Factorial and Square-Root

Below are some examples with factorial and square-root written in both ways, i.e., in digit’s order and its reverse √ √ 5040 := (5 + 0 + 4)! + 0 = 0 + ( 4 + 0 + 5)! √ √ 5041 := (5 + 0 + 4)! + 1 = 1 + ( 4 + 0 + 5)! √ √ 5042 := (5 + 0 + 4)! + 2 = 2 + ( 4 + 0 + 5)! √ √ 5043 := (5 + 0 + 4)! + 3 = 3 + ( 4 + 0 + 5)! √ √ 5044 := (5 + 0 + 4)! + 4 = 4 + ( 4 + 0 + 5)! √ √ 5045 := (5 + 0 + 4)! + 5 = 5 + ( 4 + 0 + 5)! √ √ 5046 := (5 + 0 + 4)! + 6 = 6 + ( 4 + 0 + 5)! √ √ 5047 := (5 + 0 + 4)! + 7 = 7 + ( 4 + 0 + 5)! √ √ 5048 := (5 + 0 + 4)! + 8 = 8 + ( 4 + 0 + 5)! √ √ 5049 := (5 + 0 + 4)! + 9 = 9 + ( 4 + 0 + 5)!

34

Inder J. Taneja

RGMIA Research Report Collection, 20(2017), pp. 1-42, http://rgmia.org/v20.php √ √ 936 := ( 9)!3 + 6! = 6! + (3!) 9 q √ 1296 := (1 + 2)!9 /6 = 6( 9+2−1) √ √ 2896 := 2 × (8 + ( 9)!! + 6!) = (6! + ( 9)!! + 8) × 2 √ √ 331779 := 3 + (31 − 7) 7+9 = 9 + (7 × 7 − 1)3 × 3 √ √ 4 342995 := (3 − 2 − 9) 9 − 5 = −5 + (−9 + 92 − 4)3 √

759375 := (−7 + 59 − 37)5 = (5 + 7 + 3 ) 9−5+7 . √ −3+8 759381 := 7 + (5 × 9) − 1 = −1 + (8 × 3 − 9)5 + 7. The following examples are in digit’s order and its reverse separately: 25 := 52 √ 64 := 46

120 := ((1 + 2)! − 0!)! 127 := −1 + 27 1673 := −1 − 6 + 7!/3

289 := (9 + 8)2 q √ √ 3894 := ( 4 + ( 9)!8 ) × 3

√

1679 := 1 + (−6 + 7!)/ 9 √ 1680 := (1 + 6)!/ 8 + 0! 38970 := −3!! + 8! − 9 × 70 q √ 38986 := −3 + 8! − ( 9 + 8)6 √ 40310 := ( 403 )! − 10 √ √ 90894 := −( 9)! + ((0! + 8)! + ( 9)!!)/4 √ 91560 := (( 9)! + 1)! + 5! × (6! + 0!)

4957 := 7! − 59 − 4! 6992 := 29 + 9 × 6!

√

26493 := (2 + 6)! − 4! 9 − 3 30792 := 3! × ((0 + 7)! + 92) 54476 := (5! + 4!4 − 7!)/6 √ √ 75989 := 9 × (8 − ( 9)!!) + 57

First column numbers are in digit’s order and second columns are in reverse order of digits. For details refer author’s work [18, 19, 20, 23, 24].

4.3

Fibonacci Sequence

Fibonacci sequence numbers are well known in literature. This sequence is defined as F (0) = 0,

F (1) = 1,

F ( n + 1) = F ( n ) + F ( n − 1),

n ≥ 1.

Below are examples of selfie numbers by use of Fibonacci sequence values. This we have done in different situations, such as using F (·) and F ( F (·)) in separate works. See below examples: 834660 834661 834662 834663 834664

:= := := := :=

( F (8 × 3) × F (4) + 6) × 6 + 0 = 0 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 1 = 1 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 2 = 2 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 3 = 3 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 4 = 4 + 6 × (6 + F (4) × F (3 × 8))

35


Inder J. Taneja

834665 834666 834667 834668 834669 21960 21961 21962 21963 21964 21965 21966 21967 21968 21969

34 233 630 1178 2079

:= := := := :=

:= := := := :=

( F (8 × 3) × F (4) + 6) × 6 + 5 = 5 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 6 = 6 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 7 = 7 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 8 = 8 + 6 × (6 + F (4) × F (3 × 8)) ( F (8 × 3) × F (4) + 6) × 6 + 9 = 9 + 6 × (6 + F (4) × F (3 × 8)).

:= 2 × 1 × ( F (9) + F ( F ( F (6)))) + 0 = 0 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 1 = 1 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 2 = 2 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 3 = 3 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 4 = 4 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 5 = 5 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 6 = 6 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 7 = 7 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 8 = 8 + ( F ( F ( F (6))) + F (9)) × 1 × 2 := 2 × 1 × ( F (9) + F ( F ( F (6)))) + 9 = 9 + ( F ( F ( F (6))) + F (9)) × 1 × 2.

143 986 1178 2585 12819 24297

: = −1 + F (4 × 3) := F (9) × ( F (8) + F (6)) := F (11) × F (7) + F (8) : = F (2) + F (5 + 8 + 5) := 1 + F (2 × (8 − 1)) × F (9) : = F (2 × 4) × F (2 + 9) × F (7)

= F (3 × 4) − 1 = ( F (6) + F (8)) × F (9) = F (8) + F (7) × F (11) = F (5 + 8 + 5) + F (2) = F (9) × F ((−1 + 8) × 2) + 1 = F (7) × F (9 + 2) × F (4 × 2)

39394 74997 87937 98703

:= −3 + 93 + F (9) F(4) : = −7 × 4 + F (9 + 9 + 7) : = −8 + F (7) × F (9 × 3 − 7) := 9 × ( F (8) + F (7 × 03))

= (−4 + F (9)) × 3 + F (9)3 = F (7 + 9 + 9) − 4 × 7 = F (7) × F (3 × 9 − 7) − 8 = ( F (3 × 07) + F (8)) × 9

F (3 × F (4)) F ( F (−2 + 3 × 3)) F ( F (6)) × 30 F (11) × F (7) + F (8) (−2 + F ( F (07))) × 9

4864 := F ( F (4))8 × ( F ( F (6)) − F ( F (4))) 8759 := − F (9 − 5)7 + F ( F (8)) 8849 := −9 × F ( F ( F ( F ( F (4))) − 8)) + F ( F (8)) 9349 := − F ( F (9)/F ( F (4))) + F ( F ( F (−3 + 9)))

36 143 231 377 986 1165 1596 2592 9756

: = 6 F (3) : = F (3 × 4) − 1 := F (13) − 2 := F (−7 + 7 × 3) := ( F (6) + F (8)) × F (9) := 5 × F ( F (6 × 1 + 1)) := F ( F (6) + 9) − F ( F ( F (5 − 1))) := F (2 × 9) + F (5 + F (2)) := F ( F ( F (6))) − 5 × 7 × F (9)

First three blocks are in both ways. In the last block the first column values are in digit’s order and the second columns values are in reverse order of digits. For more details see author’s [27, 28, 29]. 36

Inder J. Taneja

4.4


Triangular Numbers

Triangular numbers are very much famous in the literature of mathematics. The general formula to write these numbers is given by T (n) = 1 + 2 + 3 + ... =

n+1 = C (n + 1, 2) 2

The examples given in above subsections are with factorial, square-root, Fibonacci sequence numbers, etc. Still, one can have similar kind of results using Triangular numbers. See below some examples: 1069 1081 2887 4965 4999 99545 99546

874 := T ( T ( T (4))) − T ( T (7) + 8) := T (10) − T (6) + T ( T (9)) 0105 := 50 + T (10) := T (1 + T (08 + 1)) := T ( T ( T ( T (2)))) + T ( T (8) + T (8)) + T (7) 1155 := − T ( T (5)) + T (51 − 1) 1224 := T ( T ( T (4)) − T ( T (2))) − 2 + 1 := T (−4 + 9) + T (− T (6) + T ( T (5))) 2418 := T (81) − T (42) := 49 + T (99) 99632 := 2 + (3 + T ( T (6) + T (9))) × T (9) := T (9) + T (9) × T ( T ( T (5) − 4)) + 5 99633 := 3 + (3 + T ( T (6) + T (9))) × T (9). := T (9) + T (9) × T ( T ( T (5) − 4)) + 6.

First column values are in digit’s order and the second column values are in reverse order of digits. In consecutive sequential values we have: 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219

:= := := := := := := := := :=

T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 0 = 0 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 1 = 1 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 2 = 2 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 3 = 3 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 4 = 4 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 5 = 5 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 6 = 6 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 7 = 7 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 8 = 8 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))) − 1 + 9 = 9 − 1 + T ( T ( T ( T ( T ( T (2))))/T ( T ( T (2))))).

For more details see author’s work [34]. Due to high quantity of numbers, we worked only up to 4 digits, i.e., from 1 to 9999.

4.5

Binomial Coefficients

Binomial coefficients are well known in literature. They are given by C (m, r ) =

m! , m ≥ r ≥ 0, m, r ∈ N. r! × (m − r )!

In above subsections, we gave examples of selfie numbers with Fibonacci sequence, Triangular numbers, etc. Still, one can have similar kind results using binomial coefficients. See below

37

Inder J. Taneja


some examples written in both ways, digit’s order and reverse order of digits: √ 6435 := C (C (6, 4), 3 + 5) = C (5 × 3, 4 + 6) 15504 := C (15 + 5, 0! + 4) = C (4 × 05, 5 × 1) √ 42504 := C (4!, 2 × 50/4) = C (4!, −05 + 24) √ 54264 := C (5 + 42 , C (6, 4)) = C (4! − 6/2, ( 4 + 5)!) 74613 := C (7 × 4 − 6, 1 × 3!) = C (3! + 16, (−4 + 7)!). 12650 := C (−1 + 26, 5 − 0!) 12870 := C (1 × 2 × 8, 7 + 0!) √ 14950 := C (−1 + 4! + 9, 5 − 0!) 18564 := C (18, (5 − 6 + 4)!) √ √ 19448 := C (19 − 4, 4 + 8) √ 26334 := C (2 + C (6, 3), 3 + 4) 43758 := C (4! − 3!, 7 − 5 + 8)

28 := C (8, 2)

√ 792 := C (2 × ( 9)!, 7) √ 924 := C (4!/2, ( 9)!) 2024 := C (4!, 2 + (0 × 2)!) 4845 := C (5 × 4, 8 − 4) √ 00378 := C (C (8, 7 − 3), 0! + 0!) √ 00792 := C (2 × ( 9)!, 7 − 0! − 0!) √ 00924 := C (4!/2, 9 × (0! + 0!)).

53130 := C (53−1 , 3! − 0!). Consecutive sequential representations: 25920 25921 25922 25923 25924 25925 25926 25927 25928 25929

:= := := := := := := := := :=

(−2 + 5)!! × C (9, 2) + 0 (−2 + 5)!! × C (9, 2) + 1 (−2 + 5)!! × C (9, 2) + 2 (−2 + 5)!! × C (9, 2) + 3 (−2 + 5)!! × C (9, 2) + 4 (−2 + 5)!! × C (9, 2) + 5 (−2 + 5)!! × C (9, 2) + 6 (−2 + 5)!! × C (9, 2) + 7 (−2 + 5)!! × C (9, 2) + 8 (−2 + 5)!! × C (9, 2) + 9.

98280 := 0 + C (C (8, 2), 8 − 98281 := 1 + C (C (8, 2), 8 − 98282 := 2 + C (C (8, 2), 8 − 98283 := 3 + C (C (8, 2), 8 − 98284 := 4 + C (C (8, 2), 8 − 98285 := 5 + C (C (8, 2), 8 − 98286 := 6 + C (C (8, 2), 8 − 98287 := 7 + C (C (8, 2), 8 − 98288 := 8 + C (C (8, 2), 8 − 98289 := 9 + C (C (8, 2), 8 −

√ √ √ √ √ √ √ √ √ √

9) 9) 9) 9) 9) 9) 9) 9) 9) 9).

For more details refer author’s work [32].

4.6

S-gonal numbers

The formula for S-gonal numbers is given by P (n, s) :=

n(n − 1)(s − 2) + n, s > 2. 2

This subsection brings some examples of selfie numbrs using S-gonal numbers. These examples are in digit’s order and in reverse order of digits:

38

Inder J. Taneja


√ 8967 := 7 × P( P(6, 9), 8) √ 9504 := 4! × P( 0! + 5!, 9) √ 9744 := 4! × P(4 × 7, 9) 49281 := 1 × 8! + P(29, 4!) 49548 := −8! − P(4!, 5) + 9!/4 √ 50424 := 4! × P(−2 + 4!, 0! + 5!)

4992 := P(4!, 9 + 9 + 2) √

7744 := ( P(7, 7) − 4!) 4 √ 7896 := 7 × P(8 × 9, 6) √ 65485 := − P(6, 5) + 4 × 85 √ 65943 := P(6, 5) × (( 9)!4 − 3) 67977 := (6 + 7) × ( P(9, 7) + 7!) 72495 := − P(7 + 2, 4) + 9!/5 q √ √ 83544 := P(8, 3) × (5! − 4) 4 .

52895 := (5 + P(9, 8))2 − 5 √ 53995 := (5! − P(9, 9)) × 3!! − 5.

The consecutive sequential examples are given by √ 86640 := P(8, 6) × (6! + 4) + 0 √ 86641 := P(8, 6) × (6! + 4) + 1 √ 86642 := P(8, 6) × (6! + 4) + 2 √ 86643 := P(8, 6) × (6! + 4) + 3 √ 86644 := P(8, 6) × (6! + 4) + 4 √ 86645 := P(8, 6) × (6! + 4) + 5 √ 86646 := P(8, 6) × (6! + 4) + 6 √ 86647 := P(8, 6) × (6! + 4) + 7 √ 86648 := P(8, 6) × (6! + 4) + 8 √ 86649 := P(8, 6) × (6! + 4) + 9.

5640 5641 5642 5643 5644 5645 5646 5647 5648 5649

:= 0 + P(4!, 6) × 5 := 1 + P(4!, 6) × 5 := 2 + P(4!, 6) × 5 := 3 + P(4!, 6) × 5 := 4 + P(4!, 6) × 5 := 5 + P(4!, 6) × 5 := 6 + P(4!, 6) × 5 := 7 + P(4!, 6) × 5 := 8 + P(4!, 6) × 5 := 9 + P(4!, 6) × 5.

For more details refer author’s work [33].

4.7

Centered Polygonal Numbers

The formula for centered polygonal numbers is given by K (n, t ) :=

t n ( n − 1) + 1, t > 2. 2

Below are some examples of selfie numbers with centered polygonal numbers. These are in digit’s order and inreverse order of digits: 2883 := K (2 × 8, 8) × 3 2888 := K (2 + 8, 8) × 8 3640 := K (3!, 6) × 40 √ 14939 := −1 + (K (4!, ( 9)!) + 3) × 9 √ 14959 := (−1 + K (4!, ( 9)!) + 5) × 9

15144 := K (15, (−1 + 4)!) × 4! 15347 := (−1 + 5)! × 3!! − K (4!, 7) 15399 := K (1 × 5!/3!, 9) × 9 q √ 00938 := K ( K (8, 3!), ( 9)!) × (0! + 0!) 01051 := K (15, 010)

39

Inder J. Taneja


√

9) × (1 + 10) √ 59938 := K (8, 3!) + ( 9)!! + 95 62424 := 4! × K (2 + 4!, 2 + 6) 01199 := K (9,

63384 := 4! + (K (8, 3) + 3) × 6! 63744 := 4! × (K (4!, 7) + 3 + 6!) 63973 := K (3! + 7, 9) × K (3!, 6).

The consecutive sequential examples are given by √ √ √ √ 99360 := K (( 9)!, 9) × 3 × 6! + 0 = 0 + 6! × K (3!, 9) × 9 √ √ √ √ 99361 := K (( 9)!, 9) × 3 × 6! + 1 = 1 + 6! × K (3!, 9) × 9 √ √ √ √ 99362 := K (( 9)!, 9) × 3 × 6! + 2 = 2 + 6! × K (3!, 9) × 9 √ √ √ √ 99363 := K (( 9)!, 9) × 3 × 6! + 3 = 3 + 6! × K (3!, 9) × 9 √ √ √ √ 99364 := K (( 9)!, 9) × 3 × 6! + 4 = 4 + 6! × K (3!, 9) × 9 √ √ √ √ 99365 := K (( 9)!, 9) × 3 × 6! + 5 = 5 + 6! × K (3!, 9) × 9 √ √ √ √ 99366 := K (( 9)!, 9) × 3 × 6! + 6 = 6 + 6! × K (3!, 9) × 9 √ √ √ √ 99367 := K (( 9)!, 9) × 3 × 6! + 7 = 7 + 6! × K (3!, 9) × 9 √ √ √ √ 99368 := K (( 9)!, 9) × 3 × 6! + 8 = 8 + 6! × K (3!, 9) × 9 √ √ √ √ 99369 := K (( 9)!, 9) × 3 × 6! + 9 = 9 + 6! × K (3!, 9) × 9. For more details refer author’s work [33].

Acknowledgements The author is thankful to T.J. Eckman, Georgia, USA (email: [email protected]) in programming the script to develop these representations. Thanks also extended to Matt Parker for introducing the idea of concatenation to write the number 10958 using 1 to 9.

References [1] ABRAHAMS, M, Lots of numbers, plain and almost simple, IMPROBABLE RESEACH, http://www.improbable.com/2013/02/12/lots-of-numbers-plain-and-almost-simple/. [2] ABRAHAMS, M, Lots more numbers, deemed "crazy consecutive", IMPROBABLE RESEACH, http://www.improbable.com/2013/06/08/lots-more-numbers-deemed-crazy-sequential/ [3] NEBUS, J., Counting To 52, nebusresearch, http://nebusresearch.wordpress.com/2013/02/17/counting-to-52/. [4] NEBUS, J., Counting From 52 to 11,108, nebusresearch, http://nebusresearch.wordpress.com/2013/06/10/counting-from-52-to-11108/. [5] C. RIVEIRA, Problems & Puzzles: Puzzles, Puzzle 864. 10958, the only hole..., http://www.primepuzzles.net/puzzles/puzz_864.htm or - https://goo.gl/Tdv2AD [6] Futility Closet, The Holdout, https://www.futilitycloset.com/?s=10958 40

Inder J. Taneja


[7] Puzzling, The 10,958 Problem, https://puzzling.stackexchange.com/questions/51129/the-10-958-problem [8] Puzzling, Rendering the number 10,958 with the string 1 2 3 4 5 6 7 8 9, https://puzzling.stackexchange.com/questions/47923/rendering-the-number-10-958-with-the-string1-2-3-4-5-6-7-8-9 [9] MATHEMATICS, Associativity of concatenation, https://math.stackexchange.com/questions/2133595/associativity-of-concatenation [10] M. PARKER, The 10,958 Problem - Numberphile, Youtube video, https://www.youtube.com/watch?v=-ruC5A9EzzE [11] M. PARKER, A 10,958 Solution - Numberphile, Youtube video, https://www.youtube.com/watch?v=pasyRUj7UwM [12] B HARAN, 10958, http://www.bradyharanblog.com/blog/10958 [13] R. MULLER, Concatenation of the numbers from 1 to n, https://oeis.org/A007908. [14] J.S. MADACHY, Mathematics on Vacations, Charlers Scriber’s Son, New York, 1966. [15] H. BOTTOMLEY, Concatenate next digit at right hand end (where the next digit after 9 is again 0), https://oeis.org/A057137. [16] C. A. PICKOVER, A Passion for Mathematics, Wiley, 2005; see p. 61 - Also refer http://sprott.physics.wisc.edu/pickover/trianglegod.html [17] I.J. TANEJA, Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9, Jan. 2014, pp.1-161, http://arxiv.org/abs/1302.1479. [18] TANEJA, I.J., Selfie Numbers: Consecutive Representations in Increasing and Decreasing Orders, RGMIA Research Report Collection, 17(2014), Article 140, pp. 1-57. http://rgmia.org/papers/v17/v17a140.pdf. [19] I.J. TANEJA, Different Types of Pretty Wild Narcissistic Numbers: Selfie Representations - I, RGMIA Research Report Collection, 18(2015), Article 32, pp.1-43. http://rgmia.org/papers/v18/v18a32.pdf. [20] I.J. TANEJA, Selfie Numbers: Representations in Increasing and Decreasing Orders of Non Consecutive Digits, RGMIA Research Report Collection, 18(2015), Article 70, pp.1-104. http://rgmia.org/papers/v18/v18a70.pdf. [21] I.J. TANEJA, Unified Selfie Numbers, RGMIA Research Report Collection, 18(2015), Article 153, pp. 1-14. http://rgmia.org/papers/v18/v18a153.pdf. [22] I.J. TANEJA, Patterns in Selfie Numbers, RGMIA Research Report Collection, 18(2015), Article 154, pp. 1-41. http://rgmia.org/papers/v18/v18a154.pdf.

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[23] I.J. TANEJA, Selfie Numbers - I: Symmetrical and Unified Representations, RGMIA Research Report Collection, 18(2015), Article 174, pp.1-94. http://rgmia.org/papers/v18/v18a174.pdf. [24] I.J. TANEJA, Selfie Numbers - II: Six Digits Symmetrical, Unified and Patterned Representations Without Factorial, RGMIA Research Report Collection, 18(2015), Article 175, pp.1-41. http://rgmia.org/papers/v18/v18a175.pdf. [25] I.J. TANEJA, Selfie Numbers - III: With Factorial and Without Square-Root - Up To Five Digits, RGMIA Research Report Collection, 19(2016), Article 16, pp.1-52, http://rgmia.org/papers/v19/v19a16.pdf. [26] I.J. TANEJA, Selfie Power Representations, RGMIA Research Report Collection, 19(2016), Article 17, pp. 1-20, http://rgmia.org/papers/v19/v19a17.pdf. [27] I.J. TANEJA, Fibonacci Sequence and Selfie Numbers - I, RGMIA Research Report Collection, 19(2016), Art 142, pp. 1-59, http://rgmia.org/papers/v19/v19a142.pdf. [28] I.J. TANEJA, Fibonacci Sequence and Selfie Numbers - II, RGMIA Research Report Collection, 19(2016), Art 143, pp. 1-47, http://rgmia.org/papers/v19/v19a143.pdf. [29] I.J. TANEJA, Fibonacci Sequence and Selfie Numbers - III, RGMIA Research Report Collection, 19(2016), Art 156, pp. 1-72, http://rgmia.org/papers/v19/v19a156.pdf. [30] I.J. TANEJA, Selfie Numbers - IV: Addition, Subtraction and Factorial, RGMIA Research Report Collection, 19(2016), Article 163, pp.1-42, http://rgmia.org/papers/v19/v19a163.pdf. [31] I.J. TANEJA, Selfie Numbers - V: Six Digits Symmetrical Representations with Factorial, RGMIA Research Report Collection, 19(2016), Article 164, pp.1-60, http://rgmia.org/papers/v19/v19a164.pdf. [32] I.J. TANEJA, Selfie Numbers and Binomial Coefficients, RGMIA Research Report Collection, 20(2017), pp. 1-18, Art. 25, http://rgmia.org/papers/v20/v20a25.pdf. [33] I.J. TANEJA, S-gonal and Centered Polygonal Selfie Numbers, and Connections with Binomials Coefficients, RGMIA Research Report Collection, 20(2017), pp. 1-42, http://rgmia.org/papers/v20/v20a43.pdf. [34] I.J. TANEJA, Triangular Selfie Numbers - I, RGMIA Research Report Collection, 20(2017), pp. 1-78, http://rgmia.org/v20.php. ————————————————

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