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For instance, we shall prove that $$ p(n) = \\sum_{d|n} \\sum_{k=1}^{d} \\sum_{i_0 =1}^{\\lfloor d/k \\rfloor} \\sum_{i_1 =i_0}^{\\lfloor\\frac{d- i_0}{k-1} \\rfloor} \\sum_{i_2 =i_1}^{\\lfloor\\frac{d- i_0 - i_1}{k-2} \\rfloor} ... \\sum_{i_{k-3}=i_{k-4}}^{\\lfloor\\frac{n- i_0 - i_1-i_2- ...-i_{k-4}}{3} \\rfloor} \\sum_{c|(d,i_0,i_1,i_2,...,i_{k-3})} \\mu(c) (\\lfloor \\frac{d-i_0-"},"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":"1004.4849","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-04-27T16:14:56Z","cross_cats_sorted":[],"title_canon_sha256":"657c7b15a072cb50d83d811ecd6517be496c08da6e190c35694c4ae0a7da33c1","abstract_canon_sha256":"e5661cc8b7b5b2a5843de5ce42fac96536bb1cd2e9b1d287b7e07ac86880961a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:15.786454Z","signature_b64":"AUfjHjdj8EOSqPfbADsasC8dFMW74oMKjtnlZOuIz1xfKfHW5FMiDPxQYEZInDimYPMUC2t+tSLw/SVjMojADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cbe2020ac50a252af59a5e287307dd97511d57322d712eea044cf49cc6e946b","last_reissued_at":"2026-05-18T02:24:15.785807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:15.785807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Elementary formulas for integer partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohamed El Bachraoui","submitted_at":"2010-04-27T16:14:56Z","abstract_excerpt":"In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$ p(n) = \\sum_{d|n} \\sum_{k=1}^{d} \\sum_{i_0 =1}^{\\lfloor d/k \\rfloor} \\sum_{i_1 =i_0}^{\\lfloor\\frac{d- i_0}{k-1} \\rfloor} \\sum_{i_2 =i_1}^{\\lfloor\\frac{d- i_0 - i_1}{k-2} \\rfloor} ... \\sum_{i_{k-3}=i_{k-4}}^{\\lfloor\\frac{n- i_0 - i_1-i_2- ...-i_{k-4}}{3} \\rfloor} \\sum_{c|(d,i_0,i_1,i_2,...,i_{k-3})} \\mu(c) (\\lfloor \\frac{d-i_0-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4849","kind":"arxiv","version":1},"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":"1004.4849","created_at":"2026-05-18T02:24:15.785906+00:00"},{"alias_kind":"arxiv_version","alias_value":"1004.4849v1","created_at":"2026-05-18T02:24:15.785906+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4849","created_at":"2026-05-18T02:24:15.785906+00:00"},{"alias_kind":"pith_short_12","alias_value":"BS7CAIFMKCRF","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BS7CAIFMKCRFFL2Z","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BS7CAIFM","created_at":"2026-05-18T12:26:05.355336+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/BS7CAIFMKCRFFL2ZUXRIOMD53F","json":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F.json","graph_json":"https://pith.science/api/pith-number/BS7CAIFMKCRFFL2ZUXRIOMD53F/graph.json","events_json":"https://pith.science/api/pith-number/BS7CAIFMKCRFFL2ZUXRIOMD53F/events.json","paper":"https://pith.science/paper/BS7CAIFM"},"agent_actions":{"view_html":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F","download_json":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F.json","view_paper":"https://pith.science/paper/BS7CAIFM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1004.4849&json=true","fetch_graph":"https://pith.science/api/pith-number/BS7CAIFMKCRFFL2ZUXRIOMD53F/graph.json","fetch_events":"https://pith.science/api/pith-number/BS7CAIFMKCRFFL2ZUXRIOMD53F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F/action/storage_attestation","attest_author":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F/action/author_attestation","sign_citation":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F/action/citation_signature","submit_replication":"https://pith.science/pith/BS7CAIFMKCRFFL2ZUXRIOMD53F/action/replication_record"}},"created_at":"2026-05-18T02:24:15.785906+00:00","updated_at":"2026-05-18T02:24:15.785906+00:00"}