The overpartition function modulo small powers of 2
The main result of this note is that \(\overline{p}(n) \equiv 0 \pmod{64}\) for almost all natural numbers \(n\), where \(\overline{p}(n)\) denotes the number of overpartitions of \(n\). This result is obtained by relating \(\overline{p}(n)\) modulo \(64\) to a linear combination of the functions \(c_k(n)\), for \(1 \leq k \leq 5\), and then studying the \(2\)-adic properties of these functions. Here \(c_k(n)\) denotes the number of representations \(n = n_1^2 + \cdots + n_k^2\), where each \(n_i\) is positive. The author notes that preliminary calculations suggest that the main result may be true modulo arbitrary powers of \(2\). However, the methods of this paper ``run into serious limitations'' for powers of \(2\) beyond \(64\).
scientific article; zbMATH DE number 2117319
scientific article