Highly composite number

This article is about numbers having many divisors. For numbers factorized only to powers of 2, 3, 5 and 7 (also named 7-smooth numbers), see Smooth number.
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6

A highly composite number (also known as an antiprime) is a positive integer that has more divisors than all smaller positive integers. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are.

Ramanujan wrote a paper on highly composite numbers in 1915.[1]

The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (= 7!), as the ideal number of citizens in a city.[2] Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.[3]

  1. ^ Ramanujan, S. (1915). "Highly composite numbers" (PDF). Proc. London Math. Soc. Series 2. 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01.
  2. ^ Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre", Notices of the American Mathematical Society, 62 (2): 136–140. Kahane cites Plato's Laws, 771c.
  3. ^ Vardoulakis, Antonis; Pugh, Clive (September 2008), "Plato's hidden theorem on the distribution of primes", The Mathematical Intelligencer, 30 (3): 61–63, doi:10.1007/BF02985381.