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 z th 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 ∑ d|n (± 1)ᵈ⁺¹\,dᶻ ∑ d | n ( ± 1 ) d + 1 d z , ∑ d|n (-1)n/d+1\,dᶻ ∑ d | n ( - 1 ) n / d + 1 d z and ∑ d|n (-1)n/d+d\,dᶻ ∑ 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.