Superior Highly Composite Number

A superior highly composite number is a positive integer  for which there is an  such that

for all , where the function  counts the divisors of  (Ramanujan 1962, pp. 87 and 115). It can be shown that all superior highly composite numbers are highly composite and that the th superior highly composite number has the form , where the factors  are prime.

The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the corresponding sequence of primes  is 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ... (OEIS A000705).

REFERENCES:

Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc. 14, 347-407, 1915. Reprinted in Collected Papers (Ed. G. H. Hardy et al. ). New York: Chelsea, pp. 78-129, 1962.

Sloane, N. J. A. Sequences A000705/M0423 and A002201/M1591 in "The On-Line Encyclopedia of Integer Sequences."