Arithmetic properties of the sum of divisors

Abstract

The divisor function $\sigma (n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m\in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for ${\nu }_p(\sigma (n))$ are established. For $p=2$, these involve only the odd primes dividing $n$. These expressions are used to establish the bound ${\nu }_2(\sigma (n))\le \lceil {\log }_2(n)\rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is ${\nu }_p(\sigma (n))\le \lceil {\log }_p(n)\rceil$, with equality related to solutions of the Ljunggren-Nagell diophantine equation.