  We extend recent results of Ono and Raji, relating the number of
self-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, we
give a combinatorial explanation for the curious equality
$2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$. We also conjecture
that an equality of this shape holds if and only if $t=4$, proving the cases
$t\in\{2,3,5\}$ and giving partial results for $t>5$.
