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We extend recent results of Ono and Raji, relating the number ofself-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, wegive a combinatorial explanation for the curious equality$2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$. We also conjecturethat 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 $t>5$.

On $t$-core and self-conjugate $(2t-1)$-core partitions in arithmetic progressions