A255937 Number of distinct products of distinct factorials up to n!.

1, 1, 2, 4, 8, 16, 28, 56, 108, 204, 332, 664, 1114, 2228, 4078, 7018, 11402, 22804, 40638, 81276, 140490, 230328, 391544, 783088, 1287034, 2273676, 3903626, 6837760, 10368184, 20736368, 34081198, 68162396

Offset: 0

Author

Charles R Greathouse IV, Mar 11 2015

Keywords

• nonn
• more

Examples

			a(3) = |{1!, 2!, 3!, 2!*3!}| = |{1, 2, 6, 12}| = 4.
		

Links

• Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.

Crossrefs

Cf. A058295, A000142, A001013, A060957.

Programs

Maple

s:= proc(n) option remember; (f-> `if`(n=0, {f},
      map(x-> [x, x*f][], s(n-1))))(n!)
    end:
a:= n-> nops(s(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 16 2015

Mathematica

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]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, May 01 2022 *)

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]))

Formula

Erdős and Graham prove that log a(n) ~ n log log n/log n.

a(p) = 2*a(p-1) for prime p. - Jon E. Schoenfield, Apr 01 2015

Extensions

More terms from Alois P. Heinz, Mar 16 2015

a(31) (=2*a(30)) from Jon E. Schoenfield, Apr 01 2015