{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WLXO6H2WKQH2OP5U3QVMVQJVPG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d726897e7dd03d4ee71fbfb6e84d6e12fc90a10be42934289ad2405972c52de6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-11T12:17:49Z","title_canon_sha256":"995b63e57853b793d81d7d2a5181e266b9c6a38dec37b862a094cc0cd887eec3"},"schema_version":"1.0","source":{"id":"1303.2498","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.2498","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"arxiv_version","alias_value":"1303.2498v2","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.2498","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"pith_short_12","alias_value":"WLXO6H2WKQH2","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WLXO6H2WKQH2OP5U","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WLXO6H2W","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:0acef3d35e553423fc1554d8d36fdfa09bf015b7958b372755a6135a85cf938d","target":"graph","created_at":"2026-05-18T03:06:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Andreas Weiermann, Hans Vernaeve, Jasson Vindas","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-11T12:17:49Z","title":"Asymptotic distribution of integers with certain prime factorizations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2498","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7178bf1d1afde54c1514882681ec7b5222a6457fe7903c9ce76fb175ae172842","target":"record","created_at":"2026-05-18T03:06:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d726897e7dd03d4ee71fbfb6e84d6e12fc90a10be42934289ad2405972c52de6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-11T12:17:49Z","title_canon_sha256":"995b63e57853b793d81d7d2a5181e266b9c6a38dec37b862a094cc0cd887eec3"},"schema_version":"1.0","source":{"id":"1303.2498","kind":"arxiv","version":2}},"canonical_sha256":"b2eeef1f56540fa73fb4dc2acac13579aaa8c885f0cf0dc0f787a66c28df6b2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b2eeef1f56540fa73fb4dc2acac13579aaa8c885f0cf0dc0f787a66c28df6b2b","first_computed_at":"2026-05-18T03:06:04.943020Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:04.943020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"emoID4fR/ccgwrt1u30B1ykW/i4FBH3xXv/kecszohIq9nvVsrZd1R7lFKjYSRLEwX5xxuc2FUqb5gQhydMkCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:04.943866Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.2498","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7178bf1d1afde54c1514882681ec7b5222a6457fe7903c9ce76fb175ae172842","sha256:0acef3d35e553423fc1554d8d36fdfa09bf015b7958b372755a6135a85cf938d"],"state_sha256":"23acd920aa94bbdead81d46ee42c1a860246ef93b4b72d4079c60db1a24731e4"}