highly composite number
nounEtymology 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.
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Mathematics, Number theory, Sciences A positive integer that has more divisors than any smaller positive integer.
- In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q#95;#123;2k#125;(N),#92;1#92;lek#92;le 4, where Q#95;#123;2k#125;(N) denotes the number of representations of N as the sum of 2k squares, and to #92;sigma#95;#123;-s#125;(N), where #92;sigma#95;#123;-s#125;(N) denotes the sum of the (-s)th powers of the divisors of N.2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359:
- Hardy has stated that a highly composite number is as unlike a prime as a number can be.1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48:
2 more examples
- Ramanujan devoted a section of his paper to the study of Q(x), the number of highly composite numbers #92;lex Since d(2n)gt;d(n), we see that between x and 2x, there is always a highly composite number.2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144:
- 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.2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232:
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Used other than figuratively or idiomatically: see highly, composite number; A positive integer that has a relatively large number of divisors.
- This factorization becomes particularly simple and economical when N is a highly composite number, in particular a power of 2.1995, Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Paperback edition, Cambridge University Press, published 1998, page 176:
- However, the FFT algorithm requires that the number of input points be a highly composite number of 2ᴺ; see Rabiner and Gold (1975).2004, Roger G. Jackson, Novel Sensors and Sensing, Institute of Physics Publishing, page 275:
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- 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.2010, Kenneth Lange, Numerical Analysis for Statisticians, 2nd edition, Springer, page 395:
- Synonyms
- HCN, antiprime
- Related terms
- abundant number, superabundant number
- Hypernyms
- composite number, largely composite number
- Hyponyms
- superior highly composite number