On the unimodal character of the frequency function of the largest prime factor

• Autorzy: Jean-Marie De Koninck , Jason Pierre Sweeney
• Języki publikacji: EN

Abstrakty

EN

The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in I₂(x), in which interval it oscillates, and finally decreases for p ∈ I₃(x). In fact, we show that v(x) ≥ √(log x) and w(x) ≤ √x. We also provide several conditions on primes p ≤ q so that f(x,p) ≥ f(x,q).

Kategorie tematyczne

• 11N25: Distribution of integers with specified multiplicative constraints

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 88
• Numer: 2
• Strony: 159-174

Daty

wydano

2001

Twórcy

autor

Jean-Marie De Koninck

• Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, Canada

autor

Jason Pierre Sweeney

• Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, Canada

Identyfikatory

DOI

10.4064/cm88-2-1