"Farey sequence" meaning in English

{
  "etymology_text": "Named after British geologist John Farey Sr., whose letter about the sequences was published in the Philosophical Magazine in 1816.",
  "forms": [
    {
      "form": "Farey sequences",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Farey sequence (plural Farey sequences)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "2002, Alfred S. Posamentier, Jay Stepelman, Teaching Secondary Mathematics, Merrill, page 403:",
          "text": "Students should then see the number of fractions N, in the Farey sequence is equal to #x5C;phi(2)#x2B;#x5C;phi(3)#x2B;#x5C;phi(4)#x2B;#x5C;dots#x2B;#x5C;phi(n), where #x5C;phi(n) is the number of positive integers less than or equal to n that are relatively prime to n.",
          "type": "quote"
        },
        {
          "text": "2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, LNCS 4698, page 218,\nThe Farey sequence of order n (denoted ℱₙ) is the increasing sequence of all irreducible fractions from interval [0,1] with denominators less than or equal to n. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics."
        },
        {
          "ref": "2009, Michel Weber, Dynamical Systems and Processes, European Mathematical Society, page 549:",
          "text": "Riemann sums have also important connections with various problems from number theory, among them the Riemann Hypothesis, through their link with Farey sequences.",
          "type": "quote"
        }
      ],
      "glosses": [
        "For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size."
      ],
      "id": "en-Farey_sequence-en-noun-nbgDel~v",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "fraction",
          "fraction"
        ],
        [
          "denominator",
          "denominator"
        ]
      ],
      "raw_glosses": [
        "(number theory) For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size."
      ],
      "related": [
        {
          "word": "Farey neighbour"
        },
        {
          "word": "Farey pair"
        }
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Farey sequence",
        "John Farey Sr.",
        "Philosophical Magazine"
      ]
    }
  ],
  "word": "Farey sequence"
}

{
  "etymology_text": "Named after British geologist John Farey Sr., whose letter about the sequences was published in the Philosophical Magazine in 1816.",
  "forms": [
    {
      "form": "Farey sequences",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Farey sequence (plural Farey sequences)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Farey neighbour"
    },
    {
      "word": "Farey pair"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Number theory"
      ],
      "examples": [
        {
          "ref": "2002, Alfred S. Posamentier, Jay Stepelman, Teaching Secondary Mathematics, Merrill, page 403:",
          "text": "Students should then see the number of fractions N, in the Farey sequence is equal to #x5C;phi(2)#x2B;#x5C;phi(3)#x2B;#x5C;phi(4)#x2B;#x5C;dots#x2B;#x5C;phi(n), where #x5C;phi(n) is the number of positive integers less than or equal to n that are relatively prime to n.",
          "type": "quote"
        },
        {
          "text": "2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, LNCS 4698, page 218,\nThe Farey sequence of order n (denoted ℱₙ) is the increasing sequence of all irreducible fractions from interval [0,1] with denominators less than or equal to n. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics."
        },
        {
          "ref": "2009, Michel Weber, Dynamical Systems and Processes, European Mathematical Society, page 549:",
          "text": "Riemann sums have also important connections with various problems from number theory, among them the Riemann Hypothesis, through their link with Farey sequences.",
          "type": "quote"
        }
      ],
      "glosses": [
        "For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "fraction",
          "fraction"
        ],
        [
          "denominator",
          "denominator"
        ]
      ],
      "raw_glosses": [
        "(number theory) For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Farey sequence",
        "John Farey Sr.",
        "Philosophical Magazine"
      ]
    }
  ],
  "word": "Farey sequence"
}