From sums of divisors to partition congruences

Let z be a complex number. For any positive integer n it is well known that the sum of the zth powers of the positive divisors of n can be computed without knowing all the divisors of n, if we take into account the factorization of n. In this paper, we rely on the integer partitions of n in order to investigate computational methods for \(\sum _{d|n} (\pm 1)^{d+1}\,d^z\) ${\sum }_{d|n}{(\pm 1)}^{d+1}{d}^z$ , \(\sum _{d|n} (-1)^{n/d+1}\,d^z\) ${\sum }_{d|n}{(-1)}^{n/d+1}{d}^z$ and \(\sum _{d|n} (-1)^{n/d+d}\,d^z\) ${\sum }_{d|n}{(-1)}^{n/d+d}{d}^z$ . 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.