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We will show, that\n  $\\bullet$ $u_n\\sim3p_n$.\n  Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that\n  $\\bullet$ $f_n$ is an odd number for $n\\geq5$ and\n  $\\bullet$ $f_n\\sim3p_n$; further\n  $\\bullet$ $4p_n>f_{n+1}$ for $n\\geq1$.\n  If $f_n$ is odd for large $n$, then $f_n\\sim3p_n$. In case $f_n\\sim3p_n$ every large even integer $x$ is the sum of two","authors_text":"Anton Rechenauer, Michael Hellus, Rolf Waldi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","title":"Numerical Semigroups generated by Primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09483","kind":"arxiv","version":3},"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":"sha256:8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432","target":"record","created_at":"2026-07-05T01:08:24Z","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":"e6dc03950a039c17f3d28730bc9a25da75970fe417355e03067ea2834675f266","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-26T05:54:26Z","title_canon_sha256":"709f66c441bcaccb1c94f96434b1f642748638f7236fd6011044cb49ad651b1c"},"schema_version":"1.0","source":{"id":"1908.09483","kind":"arxiv","version":3}},"canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f26cbcee949bec96f136b6fc9932e752a1954c78f98afdb0ffe85d818abf0f7","first_computed_at":"2026-07-05T01:08:24.528139Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:24.528139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XEVlIQhviV7Ah9R5XVdP9QKxXL0YYkplFL4bkT0ygF4Egy2zA6MyKI9ufPMW+zqNX786d8VDZ8JJDf5BVVM4Cg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:24.528595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.09483","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8cda1174eef253147b52b005159bedb5e3d579ccd5b7cb2d96823904ac8d3432","sha256:ac8d0041b39ad3b092a02317066001984325f3b30e718a5ca414215108d4de27"],"state_sha256":"2c305582700fb7cb2f3706f200e7162c540d84bcf5fafc4cfaf693af6d434e43"}