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These resemble the Hecke-type congruences found by Atkin for the partition function $p(n)$ in 1966 and Garvan for the smallest parts function $\\operatorname{spt}(n)$ in 2010. The proofs depend on congruences between the generating functions for $\\bar{p}(n)$, $\\bar{\\operatorname{spt1}}(n),$ and $\\operatorname{M2spt}(n)$ and eigenfor"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.4009","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-09-18T16:12:19Z","cross_cats_sorted":[],"title_canon_sha256":"c94b06abcc5d87ae40a62dded53115832781bbd2cdfe696dda6efbe6721d09dd","abstract_canon_sha256":"1248b48ab28a5b90fcd60ae6222ef4464d79ddc382f373732afbd067f28fa78d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:00.147622Z","signature_b64":"yg0PrDUKY8UpoUvI/AP/IYC3WUVYqz7wKACAYtzzxYrTeujGEczfhFcmjfqXkYWQSLSRTD5e27XaVdI+pgl/Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42e66c02b53df0271eee05edcb1cf1b6fa13c20f9b28f46611536434f0e034d5","last_reissued_at":"2026-05-18T02:57:00.147180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:00.147180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hecke-type congruences for two smallest parts functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nickolas Andersen","submitted_at":"2012-09-18T16:12:19Z","abstract_excerpt":"We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\\bar{p}(n)$ and two smallest parts functions: $\\bar{\\operatorname{spt1}}(n)$ for overpartitions and $\\operatorname{M2spt}(n)$ for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function $p(n)$ in 1966 and Garvan for the smallest parts function $\\operatorname{spt}(n)$ in 2010. 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