highly composite number

We call n a highly composite number if d(n)>d(m) for all m<n, where d(n) is the number of divisors of n. The first several are 1, 2, 4, 6, 12, 24. The sequence is http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002182A002182 in Sloane’s OEIS.

The integer n is superior highly composite if there is an ϵ>0 such that for all mn,


The first several superior highly composite numbers are 2, 6, 12, 60, 120, 360. The sequence is http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002201A002201 in Sloane’s encyclopedia.


  • 1 L. Alaoglu and P. Erdös, On highly composite and similar numbers. Trans. Amer. Math. Soc. 56 (1944), 448–469. http://links.jstor.org/sici?sici=0002-9947%28194411%2956%3A3%3C448%3AOHCASN%3E2.0.CO%3B2-SAvailable at www.jstor.org
Title highly composite number
Canonical name HighlyCompositeNumber
Date of creation 2013-03-22 13:40:44
Last modified on 2013-03-22 13:40:44
Owner Kevin OBryant (1315)
Last modified by Kevin OBryant (1315)
Numerical id 7
Author Kevin OBryant (1315)
Entry type Definition
Classification msc 11N56
Defines superior highly composite number