highly composite number

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.

highly composite number (plural highly composite numbers)

1. (number theory) A positive integer that has more divisors than any smaller positive integer.
   • 2012, George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, page 359:

     In the unpublished section of his notebook, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to ${\displaystyle Q_{2k}(N),\ 1\leq k\leq 4}$, where ${\displaystyle Q_{2k}(N)}$ denotes the number of representations of ${\displaystyle N}$ as the sum of ${\displaystyle 2k}$ squares, and to ${\displaystyle \sigma _{-s}(N)}$, where ${\displaystyle \sigma _{-s}(N)}$ denotes the sum of the ${\displaystyle (-s)}$th powers of the divisors of ${\displaystyle N}$.

   • 1998, K. Srinivasa Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Books, page 48:

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

   • 2013, M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, page 144:

     Ramanujan devoted a section of his paper to the study of ${\displaystyle Q(x)}$, the number of highly composite numbers ${\displaystyle \leq x}$ Since ${\displaystyle d(2n)>d(n)}$, we see that between ${\displaystyle x}$ and ${\displaystyle 2x}$, there is always a highly composite number.

   • 2013, Robert Kanigel, The Man Who Knew Infinity, Simon & Schuster, page 232:

     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.

2. Used other than figuratively or idiomatically: see highly,‎ composite number; A positive integer that has a relatively large number of divisors.
   • 1995, Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Paperback edition, Cambridge University Press, published 1998, page 176:

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

   • 2004, Roger G. Jackson, Novel Sensors and Sensing, Institute of Physics Publishing, page 275:

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

   • 2010, Kenneth Lange, Numerical Analysis for Statisticians, 2nd edition, Springer, page 395:

     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.

• (positive integer with more divisors than any smaller positive integer): HCN (abbreviation), antiprime

• (positive integer with more divisors than any smaller positive integer): composite number, largely composite number

• (positive integer with more divisors than any smaller positive integer): superior highly composite number

• abundant number
• superabundant number

• Divisor function on Wikipedia.Wikipedia
• Highly totient number on Wikipedia.Wikipedia
• Superior highly composite number on Wikipedia.Wikipedia
• Divisor Function on Wolfram MathWorld
• Highly Composite Number on Wolfram MathWorld
• An Algorithm for Computing Highly Composite Numbers on the Wayback Machine