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We find many arithmetic properties of $\\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$,\n  \\[\\mathrm{pod}_{-4}\\Big({{3}^{\\alpha +1}}n+\\frac{5\\cdot {{3}^{\\alpha }}+1}{2}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\\mathrm{pod}_{-4}(n)$."},"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":"1410.5628","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-21T12:05:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0d7f74cfffbc64bb29e599951d62886d477509464af0789620a71afc382d86d9","abstract_canon_sha256":"8560682ec666f84b2447ae232909ead8cacbaea1da0bed09ff23d6f44ac0bed5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:05.799303Z","signature_b64":"Q00hQw3LtIvwlwZQoLNReLKv9rbP5YPeae3YYBlFUYf0rC5BB4hFqRY3PAiVcB1WOwZwMHN/oYszWlj67ulYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"048ee36cfbde2fe5902927d0d7c3ad16844b05ed68283410edeb25aae81d5e0b","last_reissued_at":"2026-05-18T01:28:05.798715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:05.798715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic Properties of Partition Quadruples 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":"2014-10-21T12:05:22Z","abstract_excerpt":"Let $\\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n \\ge 0$,\n  \\[\\mathrm{pod}_{-4}\\Big({{3}^{\\alpha +1}}n+\\frac{5\\cdot {{3}^{\\alpha }}+1}{2}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\\mathrm{pod}_{-4}(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5628","kind":"arxiv","version":2},"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":"1410.5628","created_at":"2026-05-18T01:28:05.798789+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.5628v2","created_at":"2026-05-18T01:28:05.798789+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5628","created_at":"2026-05-18T01:28:05.798789+00:00"},{"alias_kind":"pith_short_12","alias_value":"ASHOG3H33YX6","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"ASHOG3H33YX6LEBJ","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"ASHOG3H3","created_at":"2026-05-18T12:28:19.803747+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/ASHOG3H33YX6LEBJE7INPQ5NC2","json":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2.json","graph_json":"https://pith.science/api/pith-number/ASHOG3H33YX6LEBJE7INPQ5NC2/graph.json","events_json":"https://pith.science/api/pith-number/ASHOG3H33YX6LEBJE7INPQ5NC2/events.json","paper":"https://pith.science/paper/ASHOG3H3"},"agent_actions":{"view_html":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2","download_json":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2.json","view_paper":"https://pith.science/paper/ASHOG3H3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.5628&json=true","fetch_graph":"https://pith.science/api/pith-number/ASHOG3H33YX6LEBJE7INPQ5NC2/graph.json","fetch_events":"https://pith.science/api/pith-number/ASHOG3H33YX6LEBJE7INPQ5NC2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/action/storage_attestation","attest_author":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/action/author_attestation","sign_citation":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/action/citation_signature","submit_replication":"https://pith.science/pith/ASHOG3H33YX6LEBJE7INPQ5NC2/action/replication_record"}},"created_at":"2026-05-18T01:28:05.798789+00:00","updated_at":"2026-05-18T01:28:05.798789+00:00"}