Displaying similar documents to “Ramanujan primes: bounds, runs, twins, and gaps.”

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

Jean-Marie De Koninck, Jason Pierre Sweeney (2001)

Colloquium Mathematicae

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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, 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...