On certain sums involving the largest prime factor over integer sequences
Mihoub Bouderbala
TL;DR
The paper analyzes sums of the arithmetic function $f(n)$, defined as the least $m$ with $n\mid m!$, and leverages the relation $f(n)=P(n)$ when $P(n)^2>n$ to study their asymptotics. By decomposing the domain according to the condition $P(n)^2\le n$ or $>n$, bounding $f(n)$ in terms of $P(n)$, and applying Abel summation together with known estimates for $\sum_{n\le x} P(n)$ and the $k$-free density, the author obtains explicit main-term constants. The main results are: $\sum_{n\le x} f(n) \sim \zeta(2) x^{2}/\log x$ and, for fixed $k\ge 2$, $\sum_{n\le x, n\in S_k} f(n) \sim (\zeta^{2}(2)/(2\zeta(2k))) x^{2}/\log x$, with matching error terms $O(x^{2}/\log^{2} x)$. These findings connect factorial divisibility sums to the distribution of the largest prime factor and $k$-free densities, reflecting Alladi–Erdős-type phenomena and revealing concrete Euler product constants.
Abstract
Given an integer $ n \geq 2 $, its prime factorization is expressed as $ n = \prod p_i^{a_i} $. We define the function $ f(n) $ as the smallest positive integer satisfying the following condition: \[ ν_{p}\left(\frac{f(n)!}{n}\right) \geq 0, \quad \forall p \in \{p_1, p_2, \dots, p_s\}, \] where $ ν_{p}(m) $ denotes the $ p $-adic valuation of $ m $. The main objective of this paper is to derive an asymptotic formula for both sums $ \sum_{n\leq x} f(n) $ and $ \sum_{n \leq x, n \in S_k} f(n) $, where $ S_k $ denotes the set of all $ k $-free integers.