{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:WLXO6H2WKQH2OP5U3QVMVQJVPG","short_pith_number":"pith:WLXO6H2W","schema_version":"1.0","canonical_sha256":"b2eeef1f56540fa73fb4dc2acac13579aaa8c885f0cf0dc0f787a66c28df6b2b","source":{"kind":"arxiv","id":"1303.2498","version":2},"attestation_state":"computed","paper":{"title":"Asymptotic distribution of integers with certain prime factorizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Andreas Weiermann, Hans Vernaeve, Jasson Vindas","submitted_at":"2013-03-11T12:17:49Z","abstract_excerpt":"Let $p_{1}<p_2<... <p_{\\nu}<...$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form $p_{m^{k_1}}p_{m^{k_{2}}...p_{m^{k_{n}}}$ with $k_{1}\\le k_{2}\\le...\\le k_{n}$. Such integers originate in various combinatorial counting problems; when $m=2$, they arise as Matula numbers of certain rooted trees."},"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":"1303.2498","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-11T12:17:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"995b63e57853b793d81d7d2a5181e266b9c6a38dec37b862a094cc0cd887eec3","abstract_canon_sha256":"d726897e7dd03d4ee71fbfb6e84d6e12fc90a10be42934289ad2405972c52de6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:04.943866Z","signature_b64":"emoID4fR/ccgwrt1u30B1ykW/i4FBH3xXv/kecszohIq9nvVsrZd1R7lFKjYSRLEwX5xxuc2FUqb5gQhydMkCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2eeef1f56540fa73fb4dc2acac13579aaa8c885f0cf0dc0f787a66c28df6b2b","last_reissued_at":"2026-05-18T03:06:04.943020Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:04.943020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic distribution of integers with certain prime factorizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Andreas Weiermann, Hans Vernaeve, Jasson Vindas","submitted_at":"2013-03-11T12:17:49Z","abstract_excerpt":"Let $p_{1}<p_2<... <p_{\\nu}<...$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form $p_{m^{k_1}}p_{m^{k_{2}}...p_{m^{k_{n}}}$ with $k_{1}\\le k_{2}\\le...\\le k_{n}$. Such integers originate in various combinatorial counting problems; when $m=2$, they arise as Matula numbers of certain rooted trees."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2498","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":"1303.2498","created_at":"2026-05-18T03:06:04.943184+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.2498v2","created_at":"2026-05-18T03:06:04.943184+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.2498","created_at":"2026-05-18T03:06:04.943184+00:00"},{"alias_kind":"pith_short_12","alias_value":"WLXO6H2WKQH2","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WLXO6H2WKQH2OP5U","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WLXO6H2W","created_at":"2026-05-18T12:28:04.890932+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/WLXO6H2WKQH2OP5U3QVMVQJVPG","json":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG.json","graph_json":"https://pith.science/api/pith-number/WLXO6H2WKQH2OP5U3QVMVQJVPG/graph.json","events_json":"https://pith.science/api/pith-number/WLXO6H2WKQH2OP5U3QVMVQJVPG/events.json","paper":"https://pith.science/paper/WLXO6H2W"},"agent_actions":{"view_html":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG","download_json":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG.json","view_paper":"https://pith.science/paper/WLXO6H2W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.2498&json=true","fetch_graph":"https://pith.science/api/pith-number/WLXO6H2WKQH2OP5U3QVMVQJVPG/graph.json","fetch_events":"https://pith.science/api/pith-number/WLXO6H2WKQH2OP5U3QVMVQJVPG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG/action/storage_attestation","attest_author":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG/action/author_attestation","sign_citation":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG/action/citation_signature","submit_replication":"https://pith.science/pith/WLXO6H2WKQH2OP5U3QVMVQJVPG/action/replication_record"}},"created_at":"2026-05-18T03:06:04.943184+00:00","updated_at":"2026-05-18T03:06:04.943184+00:00"}