{"paper":{"title":"New Congruences of Partitions With Odd Parts Distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-07-21T09:44:22Z","abstract_excerpt":"Let $\\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\\ge 0$, \\[\\mathrm{pod}(3n+2)\\equiv 2{{(-1)}^{n+1}}{{r}_{5}}(8n+5) \\pmod{9}, \\] and \\[\\mathrm{pod}(5n+2)\\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3) \\pmod{5}.\\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \\in \\{11, 19\\}$ and any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have \\[\\mathrm{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5436","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}