%I #10 Jun 25 2022 21:53:54
%S 0,1,3,7,14,29,47,79,134,269,395,791,1295,2015,2687,5375,7343,14687,
%T 20519,30399,47999,95999,121439,170015,266111,338687,458639,917279,
%U 1166399,2332799,2764799,3932159,6082559,8211455,9797759,19595519
%N Number of unordered factorizations of n! into two distinct proper factors.
%F For n > 1, a(n) = A027423(n)/2 - 1. - _Ray Chandler_, Mar 07 2009
%t Table[Times@@(Last/@FactorInteger[ n! ]+1)/2-1,{n,2,40}] (* _Ray Chandler_, Mar 07 2009 *)
%o (PARI) for(k=2,40,print1(numdiv(k!)/2-1,", "))
%o (Python)
%o from math import prod
%o from collections import Counter
%o from sympy import factorint
%o def A157672(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())//2-1 # _Chai Wah Wu_, Jun 25 2022
%Y Cf. A157612.
%K nonn
%O 2,3
%A _Jaume Oliver Lafont_, Mar 04 2009, Mar 05 2009
%E Extended by _Ray Chandler_, Mar 07 2009
%E PARI program rewritten _Jaume Oliver Lafont_, Mar 09 2009