# highly composite number

We call $n$ a highly composite number if $d(n)>d(m)$ for all $m, 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 $\epsilon>0$ such that for all $m\not=n$,

 $d(n)n^{-\epsilon}>d(m)m^{-\epsilon}.$

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.

## References

• 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 HighlyCompositeNumber 2013-03-22 13:40:44 2013-03-22 13:40:44 Kevin OBryant (1315) Kevin OBryant (1315) 7 Kevin OBryant (1315) Definition msc 11N56 superior highly composite number