From sums of divisors to partition congruences

Abstract 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 ₃|₍ (1) ^d+1\, dᶻ ∑ d | n (± 1) d + 1 d z, ₃|₍ (-1) ^n/d+1\, dᶻ ∑ d | n (- 1) n / d + 1 d z and ₃|₍ (-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.