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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-1,
author = {Jean-Marie De Koninck and Jason Pierre Sweeney},
title = {On the unimodal character of the frequency function of the largest prime factor},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {159-174},
zbl = {1027.11068},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-1}
}
Jean-Marie De Koninck; Jason Pierre Sweeney. On the unimodal character of the frequency function of the largest prime factor. Colloquium Mathematicae, Tome 89 (2001) pp. 159-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-1/