Congruences for Andrews' spt-function modulo powers of 5, 7 and 13

TL;DR

This paper establishes new congruences for Andrews' spt-function modulo powers of 5, 7, and 13, extending previous prime modulus results and utilizing advanced modular form theory.

Contribution

It extends Ono's method to prove congruences for spt(n) modulo powers of 5, 7, and 13, involving weak Maass forms and modular equations.

Findings

Congruences for spt(n) modulo powers of 5, 7, and 13 are established.

The method extends Ono's prime-based approach to prime powers.

The work connects spt-function congruences with modular form theory.

Abstract

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo p for all primes p>3 which were conjectured earlier by the author. We extend Ono's method to handle the powers of 5, 7 and 13 congruences. We need the theory of weak Maass forms as well as certain classical modular equations for the Dedekind eta-function.