{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:TRTDRYJELNWPQ2RYY2CMVZ5PRG","short_pith_number":"pith:TRTDRYJE","schema_version":"1.0","canonical_sha256":"9c6638e1245b6cf86a38c684cae7af898616a584685278294da07ff738614175","source":{"kind":"arxiv","id":"2508.20853","version":1},"attestation_state":"computed","paper":{"title":"Bounds for sets of remainders","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ingrid Vukusic, Omkar Baraskar","submitted_at":"2025-08-28T14:49:24Z","abstract_excerpt":"Let $s(n)$ be the number of different remainders $n \\bmod k$, where $1 \\leq k \\leq \\lfloor n/2 \\rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \\cdot n + O(n/(\\log n \\log \\log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by $O(\\log "},"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":"2508.20853","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-08-28T14:49:24Z","cross_cats_sorted":[],"title_canon_sha256":"68a7d3e578bbcaacfaee28b30b66df33ee446a551361edc1ad9ced2f99f4287b","abstract_canon_sha256":"54ca3710f0d0465e5321cfa825f5f3c171203e2fd2c896b1573ecfc9e31aa0ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:01:12.268869Z","signature_b64":"MXge70gOM8At9TSr7XWvXwgcWOMJ+Fjf8sKnM286jxhbPZbWeVsb7Xry9vm/wg0Jlx66v6Luh3j3m/3J9w1HAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c6638e1245b6cf86a38c684cae7af898616a584685278294da07ff738614175","last_reissued_at":"2026-07-05T12:01:12.268403Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:01:12.268403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds for sets of remainders","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ingrid Vukusic, Omkar Baraskar","submitted_at":"2025-08-28T14:49:24Z","abstract_excerpt":"Let $s(n)$ be the number of different remainders $n \\bmod k$, where $1 \\leq k \\leq \\lfloor n/2 \\rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \\cdot n + O(n/(\\log n \\log \\log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by $O(\\log "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.20853","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.20853/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2508.20853","created_at":"2026-07-05T12:01:12.268460+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.20853v1","created_at":"2026-07-05T12:01:12.268460+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.20853","created_at":"2026-07-05T12:01:12.268460+00:00"},{"alias_kind":"pith_short_12","alias_value":"TRTDRYJELNWP","created_at":"2026-07-05T12:01:12.268460+00:00"},{"alias_kind":"pith_short_16","alias_value":"TRTDRYJELNWPQ2RY","created_at":"2026-07-05T12:01:12.268460+00:00"},{"alias_kind":"pith_short_8","alias_value":"TRTDRYJE","created_at":"2026-07-05T12:01:12.268460+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/TRTDRYJELNWPQ2RYY2CMVZ5PRG","json":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG.json","graph_json":"https://pith.science/api/pith-number/TRTDRYJELNWPQ2RYY2CMVZ5PRG/graph.json","events_json":"https://pith.science/api/pith-number/TRTDRYJELNWPQ2RYY2CMVZ5PRG/events.json","paper":"https://pith.science/paper/TRTDRYJE"},"agent_actions":{"view_html":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG","download_json":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG.json","view_paper":"https://pith.science/paper/TRTDRYJE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.20853&json=true","fetch_graph":"https://pith.science/api/pith-number/TRTDRYJELNWPQ2RYY2CMVZ5PRG/graph.json","fetch_events":"https://pith.science/api/pith-number/TRTDRYJELNWPQ2RYY2CMVZ5PRG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG/action/storage_attestation","attest_author":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG/action/author_attestation","sign_citation":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG/action/citation_signature","submit_replication":"https://pith.science/pith/TRTDRYJELNWPQ2RYY2CMVZ5PRG/action/replication_record"}},"created_at":"2026-07-05T12:01:12.268460+00:00","updated_at":"2026-07-05T12:01:12.268460+00:00"}