{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:OCEEX4YCMO6TEBEFUPEMCYUSOP","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":"5377885a024005337013bb20b54608abae7fc6d84d1ca11918f2efbb95b90591","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-16T07:28:45Z","title_canon_sha256":"8e15b1a71847f2fbd5255ffc8e65aea6a97b5279fb91d89f8bc8ed6762aedf34"},"schema_version":"1.0","source":{"id":"1201.3168","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3168","created_at":"2026-05-18T04:04:32Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3168v1","created_at":"2026-05-18T04:04:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3168","created_at":"2026-05-18T04:04:32Z"},{"alias_kind":"pith_short_12","alias_value":"OCEEX4YCMO6T","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OCEEX4YCMO6TEBEF","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OCEEX4YC","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:5b5013520231e0bb3c0384bfc81deff01df315ad47826a004ebe2d06606674a6","target":"graph","created_at":"2026-05-18T04:04:32Z","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":"Following Srinivasan, an integer n\\geq 1 is called practical if every natural number in [1,n] can be written as a sum of distinct divisors of n. This motivates us to define f(n) as the largest integer with the property that all of 1, 2, 3,..., f(n) can be written as a sum of distinct divisors of n. (Thus, n is practical precisely when f(n)\\geq n.) We think of f(n) as measuring the \"practicality\" of n; large values of f correspond to numbers n which we term practical pretenders. Our first theorem describes the distribution of these impostors: Uniformly for 4 \\leq y \\leq x, #{n\\leq x: f(n)\\geq y","authors_text":"Lola Thompson, Paul Pollack","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-16T07:28:45Z","title":"Practical pretenders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3168","kind":"arxiv","version":1},"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:feeb80d5b6d70ba864f66df1e9a49442697dd6f34f7b553f820a9ee06c7f830e","target":"record","created_at":"2026-05-18T04:04:32Z","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":"5377885a024005337013bb20b54608abae7fc6d84d1ca11918f2efbb95b90591","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-01-16T07:28:45Z","title_canon_sha256":"8e15b1a71847f2fbd5255ffc8e65aea6a97b5279fb91d89f8bc8ed6762aedf34"},"schema_version":"1.0","source":{"id":"1201.3168","kind":"arxiv","version":1}},"canonical_sha256":"70884bf30263bd320485a3c8c1629273c0829f6cb22a66a7b78cd0f466c37669","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70884bf30263bd320485a3c8c1629273c0829f6cb22a66a7b78cd0f466c37669","first_computed_at":"2026-05-18T04:04:32.020337Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:32.020337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y5Fj/VuhExEEJElQo7ID9zIYNQUUaI+wawOWTnAblfzcXAWHRWrR1LlAIKUPKblOxBdvn4qlxtFyuaNrK20eCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:32.020935Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3168","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:feeb80d5b6d70ba864f66df1e9a49442697dd6f34f7b553f820a9ee06c7f830e","sha256:5b5013520231e0bb3c0384bfc81deff01df315ad47826a004ebe2d06606674a6"],"state_sha256":"03758d001661eafd7e6785016967e6f9507a2a25bf38cc4fb247c564686e216c"}