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By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite famil"},"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":"1802.01374","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T13:02:33Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"5d26a2a1dad30cec1b00c7776c355a7571d87a87bcc96359ba8ad4fe47b5dfa3","abstract_canon_sha256":"8d97ab9ce69e5032318aa553f41e7b6c76dcd88437b70dcc7af9d482d8737662"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:20.214151Z","signature_b64":"gIWV7O4ET4kG504i/hioEod9TOq3yaJ3gGqqQ/qWVZOuZKcHtLLh+/6mceLEQwXGvMGPHi3u4XJIno8iVbIxBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21e87670db65034ba4e5b400ccacd663214e12674bf82c1fcbcd9ab97288669c","last_reissued_at":"2026-05-18T00:21:20.213529Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:20.213529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences for the Coefficients of the Powers of the Euler Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Edward Y.S. Liu, Jack C.D. Zhao, Julia Q.D. Du","submitted_at":"2018-02-05T13:02:33Z","abstract_excerpt":"Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\\prod _{n=1}^{\\infty}(1-q^n)^k=\\sum_{n=0}^{\\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite famil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01374","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":"1802.01374","created_at":"2026-05-18T00:21:20.213634+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.01374v2","created_at":"2026-05-18T00:21:20.213634+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01374","created_at":"2026-05-18T00:21:20.213634+00:00"},{"alias_kind":"pith_short_12","alias_value":"EHUHM4G3MUBU","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"EHUHM4G3MUBUXJHF","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"EHUHM4G3","created_at":"2026-05-18T12:32:22.470017+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/EHUHM4G3MUBUXJHFWQAMZLGWMM","json":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM.json","graph_json":"https://pith.science/api/pith-number/EHUHM4G3MUBUXJHFWQAMZLGWMM/graph.json","events_json":"https://pith.science/api/pith-number/EHUHM4G3MUBUXJHFWQAMZLGWMM/events.json","paper":"https://pith.science/paper/EHUHM4G3"},"agent_actions":{"view_html":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM","download_json":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM.json","view_paper":"https://pith.science/paper/EHUHM4G3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.01374&json=true","fetch_graph":"https://pith.science/api/pith-number/EHUHM4G3MUBUXJHFWQAMZLGWMM/graph.json","fetch_events":"https://pith.science/api/pith-number/EHUHM4G3MUBUXJHFWQAMZLGWMM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/action/storage_attestation","attest_author":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/action/author_attestation","sign_citation":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/action/citation_signature","submit_replication":"https://pith.science/pith/EHUHM4G3MUBUXJHFWQAMZLGWMM/action/replication_record"}},"created_at":"2026-05-18T00:21:20.213634+00:00","updated_at":"2026-05-18T00:21:20.213634+00:00"}