Meaning of highly composite number | Babel Free

Definitions

1. A positive integer that has more divisors than any smaller positive integer.
2. Used other than figuratively or idiomatically: see highly, composite number; A positive integer that has a relatively large number of divisors.

Equivalents

Examples

“In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number⟳ to other arithmetic functions, mainly to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.”

“Hardy has stated that a highly composite number⟳ is as unlike a prime as a number⟳ can be.”

“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⟳.”

“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.”

“This factorization becomes particularly simple and economical when N is a highly composite number⟳, in particular a power of 2.”

“However, the FFT algorithm requires that the number⟳ of input points be a highly composite number⟳ of 2ᴺ; see⟳ Rabiner and Gold (1975).”

“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.”

CEFR level