A120690 Number of ordered finite sequences a_1 <= a_2 <= ... <= a_n of length n of positive integers less than or equal to n whose product is n!.

1, 1, 1, 1, 2, 2, 6, 6, 13, 33, 97, 97, 306, 306, 896, 2804, 7974, 7974, 24929, 24929, 82612, 263913, 782310, 782310, 2692708, 7705973, 22901921, 69066986, 232517327, 232517327, 853599455, 853599455, 2737549933, 8929178758, 26621287019, 90741181490, 325528535021, 325528535021

Offset

0,5

Links

Table of n, a(n) for n=0..37.

Formula

a(p) = a(p-1) for prime p. - Alois P. Heinz, Jul 05 2022

Example

a(5) = 2 because 5! = 1*2*3*4*5 = 2*2*2*3*5.

Mathematica

a[n_] := a[n] = Module[{b}, b[c_, p_, m_] := b[c, p, m] = Module[{x}, If[c <= 0 || m <= 1 || p > m^c, Boole[p == 1], x = IntegerExponent[p, m]; Sum[b[c - i, p/m^i, m - 1], {i, x*Boole@PrimeQ[m], x}]]]; b[n, n!, n]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 37}] (* Jean-François Alcover, Jul 05 2022, after Martin Fuller *)

Prog

(PARI) a(n) = (b(c, p, m) = local(x); if(c<=0||m<=1||p>m^c, p==1, x=valuation(p, m); sum(i=x*isprime(m), x, b(c-i, p/m^i, m-1)))); b(n, n!, n)

Crossrefs

Cf. A000040, A119551.

Sequence in context: A337735 A060549 A228315 * A165124 A288302 A286409

Adjacent sequences: A120687 A120688 A120689 * A120691 A120692 A120693

Keyword

nonn

Author

Martin Fuller, Jun 26 2006

Extensions

a(0)=1 prepended and a(24)-a(37) added by Alois P. Heinz, Jul 05 2022

Status

approved