Superior Highly Composite Numbers
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&lt;=m,n++,i=Position[x,Max[x]][[1,1]];A[l,f[[i,1]]];f[[i,2]]++;If[i&gt;o,o++;A[f,{Prime[i+1],0}];A[x,p[f[[-1]]]]];x[[i]]=p[f[[i]]]];Times@@l)&amp;

input

[21]

output

6064949221531200

input

[50]

output

247899128073275948560051200231228551175691632580942972608000