p-adic Valuation of the Sum of Divisors
摘要
For any positive integer n, let σ(n) be the sum of all positive divisors of n. For any prime p and any positive integer m, let νp(m) be the largest integer α such that pα ∣ m. Recently, Amdeberhan, Moll, Sharma and Villamizar proved that for any odd prime p and any integer n ≥ 2, νp(σ(n)) ≤ [logp n] if n satisfies some conditions, where [x] denotes the least integer not less than x. In this paper, for any odd prime p, we prove that νp(σ(n)) ≤ [logp n] for all positive integers n unconditionally. Moreover, we prove that if 3 ≤ p < 105 is a prime and p ≠ 31, then there are only finitely many positive integers n such that νp(σ(n)) = [logp n]. For p = 31 and n ≥ 2, νp(σ(n)) = [logp n] if and only if n = 24, 52, 2452 and 2452(2 · 31s − 1), where s is a positive integer and 2 · 31s − 1 is a prime.