{
  "id": "1407.5436",
  "version": "v2",
  "published": "2014-07-21T09:44:22.000Z",
  "updated": "2014-10-31T12:19:49.000Z",
  "title": "New Congruences of Partitions With Odd Parts Distinct",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "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{pod}\\Big({{5}^{2\\alpha +2}}n+\\frac{a \\cdot {{5}^{2\\alpha +1}}+1}{8}\\Big)\\equiv 0 \\pmod{5}.\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-21T09:44:22.000Z",
      "abstract": "Let $\\mathrm{pod}(n)$ denotes the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ 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) \\, (\\bmod \\,9), \\] and \\[\\mathrm{pod}(5n+2)\\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3)\\, (\\bmod \\, 5).\\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: let $a = 11, 19$, for any integers $\\alpha \\ge 1$ and $n \\ge 0$, we have \\[\\mathrm{pod}\\Big({{5}^{2\\alpha +2}}n+\\frac{a \\cdot {{5}^{2\\alpha +1}}+1}{8}\\Big)\\equiv 0 \\, (\\bmod \\, 5).\\]",
      "comment": "This is an original research article about partitions. It's first completed and submitted for publication by the author in May, 2014",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-31T12:19:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "odd parts distinct",
      "partitions",
      "infinite families",
      "arithmetic relations",
      "ramanujan-type congruences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5436W"
    }
  }
}