A superior highly composite number is a positive integer n for which there is an e>0 such that (d(n))/n^e>=(d(k))/k^e for all k>1, where the function d(n) counts the divisors of n. It can be shown that all superior highly composite numbers are highly composite and that the nth superior highly composite number has the form π_1 π_2 π_3 ...π_n, where the factors π_k are prime.

abundant number | colossally abundant number | highly composite number | round number | smooth number