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

0,3

Links

Table of n, a(n) for n=0..31.

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

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

Example

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

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 *)

Prog

(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]))

Crossrefs

Cf. A058295, A000142, A001013, A060957.

Sequence in context: A326116 A054189 A127195 * A357763 A330289 A348413

Adjacent sequences: A255934 A255935 A255936 * A255938 A255939 A255940

Keyword

nonn,more

Author

Charles R Greathouse IV, Mar 11 2015

Extensions

More terms from Alois P. Heinz, Mar 16 2015

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

Status

approved