A further look at the overpartition function modulo 24 and 25.

In this paper, we describe a systematic way of obtaining the exact generating functions for p ¯ (2 n) , p ¯ (4 n) (first proved by Fortin et al.), p ¯ (8 n) , p ¯ (16 n) , etc. where p ¯ (n) denotes the number of overpartitions of n. We further establish several new infinite families of congruences modulo 2 4 and 2 5 for p ¯ (n) . For example, we prove that for all n , α , β ≥ 0 and primes p ≥ 5 , p ¯ 3 4 α + 1 p 2 β + 1 24 p n + 24 j + 7 p ≡ 0 (mod 2 5) and p ¯ 3 2 α + 1 (24 n + 23) ≡ 0 (mod 2 5) , where ( - 6 p) = - 1 and 1 ≤ j ≤ p - 1 . The last congruence was proved by Xiong (Int J Number Theory 12:1195–1208, 2016) for modulo 2 4 . [ABSTRACT FROM AUTHOR]