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Normal numbers and the middle prime factor of an integer

Jean-Marie De KoninckImre Kátai — 2014

Colloquium Mathematicae

Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De KoninckImre Kátai — 2016

Colloquium Mathematicae

We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting s q ( n ) stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ( α s q ( P ( n ) ) ) n is uniformly distributed modulo 1.

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