Arithmetic properties of Schur-type overpartitions

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.