"divisibility sequence" meaning in English

{
  "forms": [
    {
      "form": "divisibility sequences",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "divisibility sequence (plural divisibility 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": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "strong divisibility sequence"
        }
      ],
      "examples": [
        {
          "ref": "2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244",
          "text": "If #x5C;mathcalC#x3D;(C#x5F;n)#x5F;#x7B;n#x5C;ge 1#x7D; is any divisibility sequence, one says that a prime p is a primitive divisor of C#x5F;n if p#x5C;vertC#x5F;n but p#x5C;nmidC#x5F;1C#x5F;2C#x5F;#x7B;n-1#x7D;. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19ᵗʰ century.",
          "type": "quotation"
        },
        {
          "ref": "2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138",
          "text": "These divisibility sequences satisfy the same recurrence relations as the polynomials #x5C;phi#x5F;n and #x5C;psi²#x5F;n (see Appendix C).",
          "type": "quotation"
        },
        {
          "ref": "2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59",
          "text": "Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a#x5F;0 = 0 and a#x5F;1#x3D;1.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ."
      ],
      "id": "en-divisibility_sequence-en-noun-wN4O5g7k",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "algebra",
          "algebra"
        ],
        [
          "sequence",
          "sequence"
        ],
        [
          "integer",
          "integer"
        ],
        [
          "index",
          "index"
        ],
        [
          "natural number",
          "natural number"
        ],
        [
          "divisible",
          "divisible"
        ]
      ],
      "raw_glosses": [
        "(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "type of integer sequence",
          "tags": [
            "feminine"
          ],
          "word": "suite à divisibilité"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "type of integer sequence",
          "tags": [
            "feminine"
          ],
          "word": "Teilbarkeitsfolge"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "type of integer sequence",
          "tags": [
            "feminine"
          ],
          "word": "successione di divisibilità"
        }
      ],
      "wikipedia": [
        "divisibility sequence"
      ]
    }
  ],
  "word": "divisibility sequence"
}

{
  "derived": [
    {
      "word": "strong divisibility sequence"
    }
  ],
  "forms": [
    {
      "form": "divisibility sequences",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "divisibility sequence (plural divisibility sequences)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Algebra",
        "en:Number theory"
      ],
      "examples": [
        {
          "ref": "2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244",
          "text": "If #x5C;mathcalC#x3D;(C#x5F;n)#x5F;#x7B;n#x5C;ge 1#x7D; is any divisibility sequence, one says that a prime p is a primitive divisor of C#x5F;n if p#x5C;vertC#x5F;n but p#x5C;nmidC#x5F;1C#x5F;2C#x5F;#x7B;n-1#x7D;. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19ᵗʰ century.",
          "type": "quotation"
        },
        {
          "ref": "2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138",
          "text": "These divisibility sequences satisfy the same recurrence relations as the polynomials #x5C;phi#x5F;n and #x5C;psi²#x5F;n (see Appendix C).",
          "type": "quotation"
        },
        {
          "ref": "2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59",
          "text": "Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a#x5F;0 = 0 and a#x5F;1#x3D;1.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "algebra",
          "algebra"
        ],
        [
          "sequence",
          "sequence"
        ],
        [
          "integer",
          "integer"
        ],
        [
          "index",
          "index"
        ],
        [
          "natural number",
          "natural number"
        ],
        [
          "divisible",
          "divisible"
        ]
      ],
      "raw_glosses": [
        "(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "divisibility sequence"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "type of integer sequence",
      "tags": [
        "feminine"
      ],
      "word": "suite à divisibilité"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "type of integer sequence",
      "tags": [
        "feminine"
      ],
      "word": "Teilbarkeitsfolge"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "type of integer sequence",
      "tags": [
        "feminine"
      ],
      "word": "successione di divisibilità"
    }
  ],
  "word": "divisibility sequence"
}