Arithmetic properties of 7-regular partitions

Abstract

Let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for $$b_{7}(n)$$ modulo powers of 7: for $$n\ge 0$$ and $$j\ge 1$$ , $$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$ We also find some infinite families of congruences modulo 2 and 7 satisfied by $$b_{7}(n)$$ .