{
  "aliases": [
    "divisor-counting-function",
    "tau-function"
  ],
  "backlinks": null,
  "body": "\n## Definition\n\nFor a positive integer $n$, the number of divisors of $n$ is $d(n) = \\sigma_0(n) = \\# D(n)$.\nHere $\\sigma_0$ is the [[divisor function|0YEW47VJ0J]] with $k = 0$, and $D(n)$ is the [[set of divisors|0FCWA16FZE]] of $n$.\n\n## Remarks\n\nOther common notations are $\\tau(n)$ and $\\sigma_0(n)$.\nIf $n = p_1^{e_1} \\cdots p_r^{e_r}$ with distinct primes $p_i$, then $d(n) = (e_1 + 1) \\cdots (e_r + 1)$.\nFor example, $12 = 2^2 \\cdot 3$, so $d(12) = 3 \\cdot 2 = 6$.\nThe number $d(n)$ is odd if and only if $n$ is a perfect square.\n\n## Explanations\n\n### L1 human\n\nThe function $d(n)$ counts the divisors of $n$.\nA divisor of $p^e q^f$ is $p^a q^b$ with $0 \\le a \\le e$ and $0 \\le b \\le f$. So there are $(e + 1)(f + 1)$ choices.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "divisor-counting-function",
      "tau-function"
    ],
    "defines": "0VV3ZMRF88",
    "kind": "definition",
    "msc": [
      "11A25"
    ],
    "name": "Number of divisors",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Divisor_function"
      }
    ],
    "tag": "0BKJT6WG96",
    "uses": [
      "0YEW47VJ0J",
      "0FCWA16FZE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0BKJT6WG96/",
  "links": [
    {
      "field": "defines",
      "target": "0VV3ZMRF88"
    },
    {
      "field": "uses",
      "target": "0YEW47VJ0J"
    },
    {
      "field": "uses",
      "target": "0FCWA16FZE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0BKJT6WG96.md",
  "msc": [
    "11A25"
  ],
  "name": "Number of divisors",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BK/0BKJT6WG96.md",
  "status": null,
  "tag": "0BKJT6WG96",
  "uri": "m2://0BKJT6WG96",
  "url": "https://math2.tamnd.com/wiki/number-of-divisors-definition/",
  "writing": {
    "tag": "0BKJT6WG96",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}