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The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, letP(n) stand for the largest prime factor ofn. Then definef(x,p) := #{nx |P(n) =p}. Iff(x,p) is considered as a function ofp, for 2 px, the primes in the interval (2,x) belong to three intervals I1(x) = (2,v(x)),I2(x) = )v(x),w(x)( andI3(x) = (w(x),x), withv(x)<w(x), such that f(x,p) increases forp 2I1(x), reaches its maximum value inI2(x), in which interval it oscillates, and finally decreases forp 2I3(x). In fact, we show thatv(x)  p logx and w(x)  p x. We also provide several conditions on primespq so thatf(x,p) f(x,q). 1. Introduction. For each integern≥ 2, letP(n) stand for its largest prime factor. Given a fixed large numberx, for each prime numberp≤x, letf(x,p) stand for the number of integersn∈ (2,x) such thatP(n) =p,

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