Congruences like Atkin's for the partition function

Scott Ahlgren, Patrick B. Allen, Shiang Tang

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p(Q^3\ell n+\beta )\equiv 0(\mathrm{mod}\ell )$ where $\ell$ and $Q$ are…