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We find many arithmetic properties of $\\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n\\ge 0$, \\[\\mathrm{pod}_{-3}\\Big({{3}^{2\\alpha +2}}n+\\frac{23\\times {{3}^{2\\alpha +1}}+3}{8}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some arithmetic relations between $\\mathrm{pod}(n)$ and $\\mathrm{pod}_{-3}(n)$, as well as some congruences for $\\mathrm{pod}_{-3}(n)$ modulo 7 and 11."},"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":"1407.5433","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-21T09:35:12Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c9b45015656e5e335654e4cf13a41c2fc6c97132bf13276df92314f7e42f21c2","abstract_canon_sha256":"3c84538998b037a916452d708ccdb58d69c8941081dc4fc299bbd6c7a6e10f17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:05.725522Z","signature_b64":"ml9ZWTOO2aB8R0yEmsBngd+Nh+Wqp3Mn7HY4KpOXT07Ciq79+k7qAEXssQqlXb2LuatjeXeZpDWkhoNvho1wCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34253b96e8a48458ca28ce90cb322288a51ec80815050d5d80052a166cb367be","last_reissued_at":"2026-05-18T01:37:05.724948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:05.724948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic Properties of Partition Triples With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-07-21T09:35:12Z","abstract_excerpt":"Let $\\mathrm{pod}_{-3}(n)$ denote the number of partition triples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n\\ge 0$, \\[\\mathrm{pod}_{-3}\\Big({{3}^{2\\alpha +2}}n+\\frac{23\\times {{3}^{2\\alpha +1}}+3}{8}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some arithmetic relations between $\\mathrm{pod}(n)$ and $\\mathrm{pod}_{-3}(n)$, as well as some congruences for $\\mathrm{pod}_{-3}(n)$ modulo 7 and 11."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5433","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.5433","created_at":"2026-05-18T01:37:05.725049+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.5433v3","created_at":"2026-05-18T01:37:05.725049+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.5433","created_at":"2026-05-18T01:37:05.725049+00:00"},{"alias_kind":"pith_short_12","alias_value":"GQSTXFXIUSCF","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GQSTXFXIUSCFRSRI","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GQSTXFXI","created_at":"2026-05-18T12:28:30.664211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC","json":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC.json","graph_json":"https://pith.science/api/pith-number/GQSTXFXIUSCFRSRIZ2IMWMRCRC/graph.json","events_json":"https://pith.science/api/pith-number/GQSTXFXIUSCFRSRIZ2IMWMRCRC/events.json","paper":"https://pith.science/paper/GQSTXFXI"},"agent_actions":{"view_html":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC","download_json":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC.json","view_paper":"https://pith.science/paper/GQSTXFXI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.5433&json=true","fetch_graph":"https://pith.science/api/pith-number/GQSTXFXIUSCFRSRIZ2IMWMRCRC/graph.json","fetch_events":"https://pith.science/api/pith-number/GQSTXFXIUSCFRSRIZ2IMWMRCRC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC/action/storage_attestation","attest_author":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC/action/author_attestation","sign_citation":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC/action/citation_signature","submit_replication":"https://pith.science/pith/GQSTXFXIUSCFRSRIZ2IMWMRCRC/action/replication_record"}},"created_at":"2026-05-18T01:37:05.725049+00:00","updated_at":"2026-05-18T01:37:05.725049+00:00"}