See highly composite number in All languages combined, or Wiktionary
Download JSON data for highly composite number meaning in English (5.7kB)
{
"etymology_text": "Coined by Indian mathematician Srinivasa Ramanujan in 1915, although it has been suggested that Plato may have known of the concept, since he specified 5040 (a highly composite number) as the ideal number of citizens in a city.",
"forms": [
{
"form": "highly composite numbers",
"tags": [
"plural"
]
}
],
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"args": {
"head": "highly composite number"
},
"expansion": "highly composite number (plural highly composite numbers)",
"name": "en-noun"
}
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"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
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"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
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"source": "w"
},
{
"_dis": "48 52",
"kind": "other",
"name": "English entries with incorrect language header",
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"examples": [
{
"ref": "2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359",
"text": "In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q#x5F;#x7B;2k#x7D;(N),#x5C;1#x5C;lek#x5C;le 4, where Q#x5F;#x7B;2k#x7D;(N) denotes the number of representations of N as the sum of 2k squares, and to #x5C;sigma#x5F;#x7B;-s#x7D;(N), where #x5C;sigma#x5F;#x7B;-s#x7D;(N) denotes the sum of the (-s)th powers of the divisors of N.",
"type": "quotation"
},
{
"ref": "1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48",
"text": "Hardy has stated that a highly composite number is as unlike a prime as a number can be.",
"type": "quotation"
},
{
"ref": "2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144",
"text": "Ramanujan devoted a section of his paper to the study of Q(x), the number of highly composite numbers #x5C;lex Since d(2n)#x3E;d(n), we see that between x and 2x, there is always a highly composite number.",
"type": "quotation"
},
{
"ref": "2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232",
"text": "A highly composite number, then, was in Hardy's phrase \"as unlike a prime as a number can be.\" Ramanujan had explored their properties for some time; in the earliest pages of his second notebook he'd listed about a hundred highly composite numbers−the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120−searching for patterns. He found them.",
"type": "quotation"
}
],
"glosses": [
"A positive integer that has more divisors than any smaller positive integer."
],
"hypernyms": [
{
"_dis1": "63 37",
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "composite number"
},
{
"_dis1": "63 37",
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "largely composite number"
}
],
"hyponyms": [
{
"_dis1": "63 37",
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "superior highly composite number"
}
],
"id": "en-highly_composite_number-en-noun-0ix6TIxU",
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisor",
"divisor"
]
],
"raw_glosses": [
"(number theory) A positive integer that has more divisors than any smaller positive integer."
],
"synonyms": [
{
"_dis1": "63 37",
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"abbreviation"
],
"word": "HCN"
},
{
"_dis1": "63 37",
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "antiprime"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"_dis1": "63 37",
"code": "de",
"lang": "German",
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"feminine"
],
"word": "hochzusammengesetzte Zahl"
},
{
"_dis1": "63 37",
"code": "sv",
"lang": "Swedish",
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"neuter"
],
"word": "mycket sammansatt tal"
}
]
},
{
"categories": [
{
"_dis": "48 52",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
}
],
"examples": [
{
"ref": "1995, Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Paperback edition, Cambridge University Press, published 1998, page 176",
"text": "This factorization becomes particularly simple and economical when N is a highly composite number, in particular a power of 2.",
"type": "quotation"
},
{
"ref": "2004, Roger G. Jackson, Novel Sensors and Sensing, Institute of Physics Publishing, page 275",
"text": "However, the FFT algorithm requires that the number of input points be a highly composite number of 2ᴺ; see Rabiner and Gold (1975).",
"type": "quotation"
},
{
"ref": "2010, Kenneth Lange, Numerical Analysis for Statisticians, 2nd edition, Springer, page 395",
"text": "We then derive the fast Fourier transform for any highly composite number n. In many applications n is a power of 2, but this choice is hardly necessary.",
"type": "quotation"
}
],
"glosses": [
"Used other than figuratively or idiomatically: see highly, composite number; A positive integer that has a relatively large number of divisors."
],
"id": "en-highly_composite_number-en-noun-RGyNon6C",
"links": [
[
"highly",
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],
[
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"composite number#English"
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[
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],
[
"integer",
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],
[
"divisor",
"divisor"
]
],
"related": [
{
"_dis1": "31 69",
"word": "abundant number"
},
{
"_dis1": "31 69",
"word": "superabundant number"
}
]
}
],
"wikipedia": [
"Plato",
"Srinivasa Ramanujan",
"highly composite number"
],
"word": "highly composite number"
}
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns"
],
"etymology_text": "Coined by Indian mathematician Srinivasa Ramanujan in 1915, although it has been suggested that Plato may have known of the concept, since he specified 5040 (a highly composite number) as the ideal number of citizens in a city.",
"forms": [
{
"form": "highly composite numbers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"head": "highly composite number"
},
"expansion": "highly composite number (plural highly composite numbers)",
"name": "en-noun"
}
],
"hypernyms": [
{
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "composite number"
},
{
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "largely composite number"
}
],
"hyponyms": [
{
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "superior highly composite number"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "abundant number"
},
{
"word": "superabundant number"
}
],
"senses": [
{
"categories": [
"English terms with quotations",
"en:Number theory"
],
"examples": [
{
"ref": "2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359",
"text": "In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q#x5F;#x7B;2k#x7D;(N),#x5C;1#x5C;lek#x5C;le 4, where Q#x5F;#x7B;2k#x7D;(N) denotes the number of representations of N as the sum of 2k squares, and to #x5C;sigma#x5F;#x7B;-s#x7D;(N), where #x5C;sigma#x5F;#x7B;-s#x7D;(N) denotes the sum of the (-s)th powers of the divisors of N.",
"type": "quotation"
},
{
"ref": "1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48",
"text": "Hardy has stated that a highly composite number is as unlike a prime as a number can be.",
"type": "quotation"
},
{
"ref": "2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144",
"text": "Ramanujan devoted a section of his paper to the study of Q(x), the number of highly composite numbers #x5C;lex Since d(2n)#x3E;d(n), we see that between x and 2x, there is always a highly composite number.",
"type": "quotation"
},
{
"ref": "2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232",
"text": "A highly composite number, then, was in Hardy's phrase \"as unlike a prime as a number can be.\" Ramanujan had explored their properties for some time; in the earliest pages of his second notebook he'd listed about a hundred highly composite numbers−the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120−searching for patterns. He found them.",
"type": "quotation"
}
],
"glosses": [
"A positive integer that has more divisors than any smaller positive integer."
],
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisor",
"divisor"
]
],
"raw_glosses": [
"(number theory) A positive integer that has more divisors than any smaller positive integer."
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
},
{
"categories": [
"English terms with quotations"
],
"examples": [
{
"ref": "1995, Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Paperback edition, Cambridge University Press, published 1998, page 176",
"text": "This factorization becomes particularly simple and economical when N is a highly composite number, in particular a power of 2.",
"type": "quotation"
},
{
"ref": "2004, Roger G. Jackson, Novel Sensors and Sensing, Institute of Physics Publishing, page 275",
"text": "However, the FFT algorithm requires that the number of input points be a highly composite number of 2ᴺ; see Rabiner and Gold (1975).",
"type": "quotation"
},
{
"ref": "2010, Kenneth Lange, Numerical Analysis for Statisticians, 2nd edition, Springer, page 395",
"text": "We then derive the fast Fourier transform for any highly composite number n. In many applications n is a power of 2, but this choice is hardly necessary.",
"type": "quotation"
}
],
"glosses": [
"Used other than figuratively or idiomatically: see highly, composite number; A positive integer that has a relatively large number of divisors."
],
"links": [
[
"highly",
"highly#English"
],
[
"composite number",
"composite number#English"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisor",
"divisor"
]
]
}
],
"synonyms": [
{
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"abbreviation"
],
"word": "HCN"
},
{
"sense": "positive integer with more divisors than any smaller positive integer",
"word": "antiprime"
}
],
"translations": [
{
"code": "de",
"lang": "German",
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"feminine"
],
"word": "hochzusammengesetzte Zahl"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "positive integer with more divisors than any smaller positive integer",
"tags": [
"neuter"
],
"word": "mycket sammansatt tal"
}
],
"wikipedia": [
"Plato",
"Srinivasa Ramanujan",
"highly composite number"
],
"word": "highly composite number"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-04-26 from the enwiktionary dump dated 2024-04-21 using wiktextract (93a6c53 and 21a9316). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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