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For example, if $p\\equiv1\\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\\equiv1\\pmod 4$ and $y\\equiv0\\pmod 2$), then $$R_{(p-1)/2}\\equiv p-(-1)^{(p-1)/4}2x\\pmod{p^2}.$$ Also, $$\\frac1{n^2}\\sum_{k=0}^{n-1}S_k\\in\\mathbb Z\\ \\ {and}\\ \\ \\frac1n\\sum_{k=0}^{n-1}S_k(x)\\in\\mathbb Z[x]\\quad\\text{for all}\\ n=1,2,3,...,$$ where $S_k(x)=\\s"},"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":"1408.5381","kind":"arxiv","version":10},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-21T16:15:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"496ee05a089a81e1e4400161739631fe4efe53eeb395c6bec553b284656afe27","abstract_canon_sha256":"da4086f8a1e247b3c5f4fa46eea958d599dfcf57a435640b8c83f6745f1aaf60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:10.936090Z","signature_b64":"Qq0w6TDEgOsy/KB7f9j4rpsOiByZ36Rm22jRICceIgEZi7WO0ehgfIHF0NWmuh5DfVhsyg4HDcmJJsF+cXljCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0f2a052ecb3742d7961750730df34b1a429e90f1510d3aaae2c8f675b76bbb2","last_reissued_at":"2026-05-18T00:01:10.935411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:10.935411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two new kinds of numbers and related divisibility results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-08-21T16:15:50Z","abstract_excerpt":"We mainly introduce two new kinds of numbers given by $$R_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k\\frac1{2k-1}\\quad\\ (n=0,1,2,...)$$ and $$S_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k(2k+1)\\quad\\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\\equiv1\\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\\equiv1\\pmod 4$ and $y\\equiv0\\pmod 2$), then $$R_{(p-1)/2}\\equiv p-(-1)^{(p-1)/4}2x\\pmod{p^2}.$$ Also, $$\\frac1{n^2}\\sum_{k=0}^{n-1}S_k\\in\\mathbb Z\\ \\ {and}\\ \\ \\frac1n\\sum_{k=0}^{n-1}S_k(x)\\in\\mathbb Z[x]\\quad\\text{for all}\\ n=1,2,3,...,$$ where $S_k(x)=\\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5381","kind":"arxiv","version":10},"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":"1408.5381","created_at":"2026-05-18T00:01:10.935512+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.5381v10","created_at":"2026-05-18T00:01:10.935512+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5381","created_at":"2026-05-18T00:01:10.935512+00:00"},{"alias_kind":"pith_short_12","alias_value":"WDZKAUXMWN2C","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WDZKAUXMWN2C26LB","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WDZKAUXM","created_at":"2026-05-18T12:28:54.890064+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/WDZKAUXMWN2C26LBOUDTBXZUWG","json":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG.json","graph_json":"https://pith.science/api/pith-number/WDZKAUXMWN2C26LBOUDTBXZUWG/graph.json","events_json":"https://pith.science/api/pith-number/WDZKAUXMWN2C26LBOUDTBXZUWG/events.json","paper":"https://pith.science/paper/WDZKAUXM"},"agent_actions":{"view_html":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG","download_json":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG.json","view_paper":"https://pith.science/paper/WDZKAUXM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.5381&json=true","fetch_graph":"https://pith.science/api/pith-number/WDZKAUXMWN2C26LBOUDTBXZUWG/graph.json","fetch_events":"https://pith.science/api/pith-number/WDZKAUXMWN2C26LBOUDTBXZUWG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/action/storage_attestation","attest_author":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/action/author_attestation","sign_citation":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/action/citation_signature","submit_replication":"https://pith.science/pith/WDZKAUXMWN2C26LBOUDTBXZUWG/action/replication_record"}},"created_at":"2026-05-18T00:01:10.935512+00:00","updated_at":"2026-05-18T00:01:10.935512+00:00"}