HighlyCompositeNumber 2003-06-11 20:25:45 2006-12-08 08:11:39 Definition superior 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 \PMlinkexternal{A002182}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002182} in Sloane's OEIS. The integer $n$ is {\em 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 \PMlinkexternal{A002201}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002201} in Sloane's encyclopedia. \begin{thebibliography}{9} \bibitem[1]{AEr1944} L. Alaoglu\ and\ P. Erd\"os, {\em On highly composite and similar numbers}. Trans. Amer. Math. Soc. {\bf 56} (1944), 448--469. \PMlinkexternal{Available at www.jstor.org}{http://links.jstor.org/sici?sici=0002-9947\%28194411\%2956\%3A3\%3C448\%3AOHCASN\%3E2.0.CO\%3B2-S} \end{thebibliography}