"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": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
          ],
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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 entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "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"
}