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%I #28 May 01 2022 13:46:10
%S 1,1,2,4,8,16,28,56,108,204,332,664,1114,2228,4078,7018,11402,22804,
%T 40638,81276,140490,230328,391544,783088,1287034,2273676,3903626,
%U 6837760,10368184,20736368,34081198,68162396
%N Number of distinct products of distinct factorials up to n!.
%H Paul Erdős and Ron L. Graham, <a href="http://www.renyi.hu/~p_erdos/1976-25.pdf">On products of factorials</a>, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.
%F Erdős and Graham prove that log a(n) ~ n log log n/log n.
%F a(p) = 2*a(p-1) for prime p. - _Jon E. Schoenfield_, Apr 01 2015
%e a(3) = |{1!, 2!, 3!, 2!*3!}| = |{1, 2, 6, 12}| = 4.
%p s:= proc(n) option remember; (f-> `if`(n=0, {f},
%p map(x-> [x, x*f][], s(n-1))))(n!)
%p end:
%p a:= n-> nops(s(n)):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 16 2015
%t a[n_] := a[n] = If[n == 0, 1, If[PrimeQ[n], 2 a[n-1], Times @@@ ((Subsets[Range[n]] // Rest) /. k_Integer -> k!) // Union // Length]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* _Jean-François Alcover_, May 01 2022 *)
%o (PARI) a(n)=my(v=[1],N=n!); for(k=2,n-1, v=Set(concat(v,v*k!))); #v + sum(i=1,#v, !setsearch(v,N*v[i]))
%Y Cf. A058295, A000142, A001013, A060957.
%K nonn,more
%O 0,3
%A _Charles R Greathouse IV_, Mar 11 2015
%E More terms from _Alois P. Heinz_, Mar 16 2015
%E a(31) (=2*a(30)) from _Jon E. Schoenfield_, Apr 01 2015
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