From sums of divisors to partition congruences (Q6552646)

The study of the sum of the \(z\)th powers of the positive divisors of the positive integer \(n\), where \(z\) is a complex number, goes back to Glaisher in the \(19\)th century.\N\NFrom the author's abstract: ``In this paper, we rely on the integer partitions of \(n\) in order to ivestigate computational methods for \(\sum_{d|n}(\pm 1)^{d+1}d^z\), \(\sum_{d|n}(-1)^{n/d+1}d^z\) and \(\sum_{d|n}(-1)^{n/d+d}d^z\). To compute these sums of divisors of \(n\), it is sufficient to know the multiplicity of \(1\) in each partition involved in the computational process. Our methods do not require knowing the divisors of \(n\) or the factorization of \(n\). New congruences involving Euler's partition function \(p(n)\) are experimentally discovered in this context.''