Abstract
The divisor function σ(n) denotes the sum of the divisors of the positive integer n. For a prime p and m∈N, the p-adic valuation of m is the highest power of p which divides m. Formulas for νp(σ(n)) are established. For p=2, these involve only the odd primes dividing n. These expressions are used to establish the bound ν2(σ(n))≤⌈log2(n)⌉, with equality if and only if n is the product of distinct Mersenne primes, and for an odd prime p, the bound is νp(σ(n))≤⌈logp(n)⌉, with equality related to solutions of the Ljunggren-Nagell diophantine equation.
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