See primorial in All languages combined, or Wiktionary
{ "etymology_templates": [ { "args": { "1": "en", "2": "prime", "3": "factorial" }, "expansion": "Blend of prime + factorial", "name": "blend" } ], "etymology_text": "Blend of prime + factorial. Coined by American engineer and mathematician Harvey Dubner.", "forms": [ { "form": "primorials", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "primorial (plural primorials)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "_dis1": "0 0", "word": "primorial prime" } ], "senses": [ { "categories": [ { "kind": "topical", "langcode": "en", "name": "Number theory", "orig": "en:Number theory", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "examples": [ { "ref": "2017, Kevin Broughan, Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents, Cambridge University Press, page 97:", "text": "For example, up to 10⁹, every positive integer which is a multiple of a primorial and is less than the following primorial satisfies the given inequality. There has also been little discussion on which primorials qualify in the different situations when RH fails.", "type": "quote" } ], "glosses": [ "Any number belonging to the integer sequence whose nth element is the product of the first n primes." ], "id": "en-primorial-en-noun-2MzgC-zZ", "links": [ [ "number theory", "number theory" ], [ "sequence", "sequence" ], [ "prime", "prime" ] ], "raw_glosses": [ "(number theory) Any number belonging to the integer sequence whose nth element is the product of the first n primes." ], "synonyms": [ { "word": "primorial number" } ], "topics": [ "mathematics", "number-theory", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Number theory", "orig": "en:Number theory", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "31 69", "kind": "other", "name": "English blends", "parents": [], "source": "w+disamb" }, { "_dis": "18 82", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "10 90", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "15 85", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "20 80", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "20 80", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "18 82", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" }, { "_dis": "18 82", "kind": "other", "name": "Terms with Turkish translations", "parents": [], "source": "w+disamb" } ], "examples": [ { "text": "The primorial of 6 is #92;textstyle 6#92;#35;#61;2#92;times 3#92;times 5#61;30.", "type": "example" }, { "text": "2020, Rong Pan, Qinheping Hu, Rishabh Singh, Loris D'Antoni, Solving Problem Sketches with Large Integer Values, Peter Müller (editor), Programming Languages and Systems: 29th European Symposium, Proceedings, Springer, LNCS 12075, page 587,\nThe following number theory result relates the primorial to the Chebyshev function.\nθ(n)= log (n#)= log 2⁽¹⁺ᵒ⁽ⁿ⁾⁾ⁿ=(1+o(n))n" } ], "glosses": [ "A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input." ], "id": "en-primorial-en-noun-6X1rFs5i", "links": [ [ "number theory", "number theory" ], [ "unary operation", "unary operation" ], [ "postfix", "postfix" ], [ "nonnegative", "nonnegative" ], [ "integers", "integers" ], [ "less than", "less than" ], [ "equal", "equal" ] ], "raw_glosses": [ "(number theory) A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input." ], "topics": [ "mathematics", "number-theory", "sciences" ] } ], "translations": [ { "_dis1": "53 47", "code": "de", "lang": "German", "sense": "product of primes", "tags": [ "feminine" ], "word": "Primorial" }, { "_dis1": "53 47", "code": "de", "lang": "German", "sense": "product of primes", "tags": [ "feminine", "rare" ], "word": "Primfakultät" }, { "_dis1": "53 47", "code": "it", "lang": "Italian", "sense": "product of primes", "tags": [ "masculine" ], "word": "primoriale" }, { "_dis1": "53 47", "code": "tr", "lang": "Turkish", "sense": "product of primes", "word": "primoriyel" } ], "wikipedia": [ "Harvey Dubner", "primorial" ], "word": "primorial" }
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