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).
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."
for which there is an 
such that
,
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.
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).
