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For each $m \\geq 3$, write $p_{m-1} + 1 = L_m R_m$ with $L_m$ the largest prime factor. Restricting to those $m$ for which $L_m > m$ (equivalently, $m \\in \\text{A223881}$), we obtain a multiset of values $R_m$. Since $p_{m-1}+1$ is even and $L_m > 3$ is odd, all values of $R_m$ are strictly even. Sorting the distinct $R_m$ by decreasing frequency yields a new sequence beginning $2, 6, 4, 8, 10, 12, 14, 16 \\dots$. This article explains how this construction arises naturally from","authors_text":"Alexander R Povolotsky","cross_cats":[],"headline":"Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-07T20:41:46Z","title":"Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.08256","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-12T01:35:55.043285Z","id":"0e0550cd-f2ad-411f-a3dc-a8ff87b9aba5","model_set":{"reader":"grok-4.3"},"one_line_summary":"Frequency-ordering the smaller factors R_m in p_{m-1}+1 factorizations for m where the largest factor exceeds m produces a new sequence explained by a heuristic model using the distribution of primes in arithmetic progressions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions.","strongest_claim":"We propose a heuristic asymptotic model explaining the observed frequency ordering via classical results on primes in arithmetic progressions and support the model with numerical log-log analysis.","weakest_assumption":"That the observed frequencies of the ratio families R_m are asymptotically governed by the distribution of primes in arithmetic progressions in a manner that can be captured by a heuristic model without post-hoc parameter tuning to the specific data set."}},"verdict_id":"0e0550cd-f2ad-411f-a3dc-a8ff87b9aba5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:db36c41cf77ac011c0d456a703f918cd0bfb9338a073df8cd5bfe91130376664","target":"record","created_at":"2026-05-28T01:04:08Z","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":"641880a4a7dd8741e8a3951676a245f339c9ea9d1ab3e24989a943161ea71d9f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-07T20:41:46Z","title_canon_sha256":"c6ac11d844b403c2e10ecb1c2261c9101da73c842e76f43e73bcba80bea0fb23"},"schema_version":"1.0","source":{"id":"2605.08256","kind":"arxiv","version":2}},"canonical_sha256":"8ec27155b629649b478874997581c45f3e69ca3bd4c4a3f1c60efbd7d371b817","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ec27155b629649b478874997581c45f3e69ca3bd4c4a3f1c60efbd7d371b817","first_computed_at":"2026-05-28T01:04:08.636144Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T01:04:08.636144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g9qf1Q/9fN1wGClOTHY0tJyzFb/YDxBffwkriugP30g1U4iJRuq0G1IeI55bE9c7p96Pv2I0pZmH6Qqx816nAQ==","signature_status":"signed_v1","signed_at":"2026-05-28T01:04:08.636987Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.08256","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db36c41cf77ac011c0d456a703f918cd0bfb9338a073df8cd5bfe91130376664","sha256:5ae5fa5094b904ea8b5b2b445227d31dcd179afb638eeee2e9391f73250d0868"],"state_sha256":"30f8dc9d0ceb0cbcdb6ef79ab4698456b3643d48b94158ed45fe5a47fb3fa855"}