On certain sums involving the largest prime factor over integer sequences

2025/04/16 by Bouderbala, Mihoub
#11A05 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.12435

Abstract

Given an integer n ≥ 2 , its prime factorization is expressed as n = ∏ p_i^a_i . We define the function f(n) as the smallest positive integer satisfying the following condition: ν_p((f(n)!)/(n)) ≥ 0, ∀ p ∈ \p₁, p₂, …, 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 ∑_{n≤ x} f(n) and ∑_{n ≤ x, n ∈ Sk} f(n) , where S_k denotes the set of all k -free integers.

Citations

• Approximate formulas for some functions of prime numbers
• Introduction to Analytic Number Theory

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