Superior Highly Composite Numbers

Question

A superior highly composite number is an integer where the ratio of its count of divisors to some power of the number is as high as possible. Expressing this as a formula:

Let d(n) be the number of divisors of n, including the number itself. For a given integer n, if there exists a number e such that d(n)/n^e is greater than or equal to d(k)/k^e for every integer k, then n is a highly composite number.

For more, see Superior highly composite number at Wikipedia, or A002201 at OEIS.

Here are the initial values:

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200

Your challenge is to take an index n, and output the nth number in this sequence.

You may use 0 or 1 indexing, and you may make a program which is only correct up to the limits of your language's data type(s), as long as it can handle the first 10 values at a minimum.

This is code golf. Standard loopholes apply.

Answer 1

Mathematica, 277 bytes

(A=AppendTo;p[f_]:=Module[{p=f[[1]],k=f[[2]]},N[Log[(k+2)/(k+1)]/Log[p]]];m=#;f={{2,1},{3,0}};o=1;l={2};x=Table[p[f[[i]]],{i,o+1}];For[n=2,n<=m,n++,i=Position[x,Max[x]][[1,1]];A[l,f[[i,1]]];f[[i,2]]++;If[i>o,o++;A[f,{Prime[i+1],0}];A[x,p[f[[-1]]]]];x[[i]]=p[f[[i]]]];Times@@l)&

input

[21]

output

6064949221531200

input

[50]

output

247899128073275948560051200231228551175691632580942972608000