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Arithmetic properties of Schur-type overpartitions

### Isaac A. Broudy and Jeremy Lovejoy

Vol. 15 (2022), No. 3, 489–505
##### Abstract

We investigate arithmetic properties of an overpartition counting function that first arose in connection with Schur’s partition theorem and a universal mock theta function. Motivated by work of Basil Gordon on the Rogers–Ramanujan identities, we first give a complete characterization of the parity of this overpartition function in the progressions $2n+1$, $4n+2$, and $8n+4$ in terms of the factorization of $An+B$ for certain $A$ and $B$. We then find similar characterizations of the residue modulo $4$ in the progressions $8n+5$ and $8n+7$. Finally, we prove some Ramanujan-type congruences modulo $5$. Our proofs use basic facts about modular forms and some elementary algebraic number theory.

##### Keywords
partitions, overpartitions, parity, eta-quotients, binary quadratic forms
##### Mathematical Subject Classification
Primary: 11P83
Secondary: 11E25, 11P84