Congruences modulo powers of 2 for Overpartition pairs into odd parts

J. Korean Math. Soc. Published online June 12, 2019

Zakir Ahmed, Rupam Barman, and Chiranjit Ray
Barnagar College, Indian Institute of Technology Guwahati

Abstract : We find congruences modulo $32$, $64$ and $128$ for the partition function $\overline{pp}_o(n)$, the number of overpartition pairs of $n$ into odd parts, with the aid of Ramamnujan's theta function identities and some known identities of $t_k(n)$, for $k=6, 7$, where $t_k(n)$ denotes the number of representations of $n$ as a sum of $k$ triangular numbers. We also find two Ramanujan-like congruences for $\overline{pp}_o(n)$ modulo $128$.