International Mathematical Forum, Vol. 7, 2012, no. 13, 615 - 620
A Note on the Distribution of Numbers with a Maximum (Minimum) Fixed Prime Factor Rafael Jakimczuk Divisi´on Matem´atica, Universidad Nacional de Luj´an Buenos Aires, Argentina
[email protected] Abstract In this note we study the distribution of numbers with a maximum (minimum) fixed prime factor in their prime factorization.
Mathematics Subject Classification: 11A99, 11B99 Keywords: Numbers with a maximum (minimum) prime factor fixed
1
Preliminary Results
Let us consider the sequence of all positive integers whose factorization is of the form q1s1 . . . qksk where si ≥ 0 (i = 1, 2, . . . , k) and q1 , . . . qk (k ≥ 2) are distinct primes fixed. Let ψ(x) denote the number of these integers not exceeding x. We need the following theorem. Theorem 1.1 The following formula holds ψ(x) =
1 lnk x 1 ln q1 + · · · + ln qk k−1 + ln x + o(lnk−1 x). (1) k! ln q1 · · · ln qk 2 (k − 1)! ln q1 · · · ln qk
Proof. See [2]. The following theorem is sometimes called either the principle of inclusionexclusion or the principle of cross-classification. We now enunciate the principle. Theorem 1.2 Let S be a set of N distinct elements, and let S1 , . . . , Sr be arbitrary subsets of S containing N1 , . . . , Nr elements, respectively. For 1 ≤ i < j < . . . < l ≤ r, let Sij...l be the intersection of Si , Sj , . . . , Sl and let
R. Jakimczuk
616
Nij...l be the number of elements of Sij...l . Then the number K of elements of S not in any of S1 , . . . , Sr is K=N−
1≤i≤r
Ni +
In particular if S = N=
1≤i≤r
r
Ni −
1≤i
