{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:HWLPT7TYCHMGQLB7N6EJRCWMHD","short_pith_number":"pith:HWLPT7TY","schema_version":"1.0","canonical_sha256":"3d96f9fe7811d8682c3f6f88988acc38e7a7006fd01fcaa78b8c4ecd232c8bdc","source":{"kind":"arxiv","id":"1601.03081","version":1},"attestation_state":"computed","paper":{"title":"The Biharmonic mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Marco Abrate, Nadir Murru, Stefano Barbero, Umberto Cerruti","submitted_at":"2016-01-12T22:01:12Z","abstract_excerpt":"We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we call the biharmonic mean. The biharmonic mean allows to introduce the biharmonic numbers, providing a new characterization for primes. Moreover, we highlight some interesting divisibility properties and we characterize the semi--prime biharmonic numbers showing their relationship with linear recurrent sequences that solve certain Diophantine equations."},"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":"1601.03081","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-12T22:01:12Z","cross_cats_sorted":[],"title_canon_sha256":"36198fd2e80cf51446d2375669dd6ad4e07e1ab3cecbcb7da7b0aa72d70cd096","abstract_canon_sha256":"f62ad9fd74bba343e75f8e63c89288f35b3e9a282082e6070a8eb13f9c343ac3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:56.934930Z","signature_b64":"Nwwv2312s4lYRCthnjMzPN1C/xWMs+8AGqOXb+wMWqF9raT525Snl73S6TUmVC0Dfczvn0YZJo0Mc8ih5oG+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d96f9fe7811d8682c3f6f88988acc38e7a7006fd01fcaa78b8c4ecd232c8bdc","last_reissued_at":"2026-05-18T01:22:56.934474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:56.934474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Biharmonic mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Marco Abrate, Nadir Murru, Stefano Barbero, Umberto Cerruti","submitted_at":"2016-01-12T22:01:12Z","abstract_excerpt":"We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we call the biharmonic mean. The biharmonic mean allows to introduce the biharmonic numbers, providing a new characterization for primes. Moreover, we highlight some interesting divisibility properties and we characterize the semi--prime biharmonic numbers showing their relationship with linear recurrent sequences that solve certain Diophantine equations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03081","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":"1601.03081","created_at":"2026-05-18T01:22:56.934554+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.03081v1","created_at":"2026-05-18T01:22:56.934554+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03081","created_at":"2026-05-18T01:22:56.934554+00:00"},{"alias_kind":"pith_short_12","alias_value":"HWLPT7TYCHMG","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"HWLPT7TYCHMGQLB7","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"HWLPT7TY","created_at":"2026-05-18T12:30:22.444734+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/HWLPT7TYCHMGQLB7N6EJRCWMHD","json":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD.json","graph_json":"https://pith.science/api/pith-number/HWLPT7TYCHMGQLB7N6EJRCWMHD/graph.json","events_json":"https://pith.science/api/pith-number/HWLPT7TYCHMGQLB7N6EJRCWMHD/events.json","paper":"https://pith.science/paper/HWLPT7TY"},"agent_actions":{"view_html":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD","download_json":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD.json","view_paper":"https://pith.science/paper/HWLPT7TY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.03081&json=true","fetch_graph":"https://pith.science/api/pith-number/HWLPT7TYCHMGQLB7N6EJRCWMHD/graph.json","fetch_events":"https://pith.science/api/pith-number/HWLPT7TYCHMGQLB7N6EJRCWMHD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD/action/storage_attestation","attest_author":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD/action/author_attestation","sign_citation":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD/action/citation_signature","submit_replication":"https://pith.science/pith/HWLPT7TYCHMGQLB7N6EJRCWMHD/action/replication_record"}},"created_at":"2026-05-18T01:22:56.934554+00:00","updated_at":"2026-05-18T01:22:56.934554+00:00"}