A316784 Number of orderless identity tree-factorizations of n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 10, 1, 2, 2, 4, 1, 8, 1, 6, 2, 2, 2, 13, 1, 2, 2, 10, 1, 8, 1, 4, 4, 2, 1, 26, 1, 4, 2, 4, 1, 10, 2, 10, 2, 2, 1, 28, 1, 2, 4, 13, 2, 8, 1, 4, 2, 8, 1, 46, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 1

Offset: 1

Author

Gus Wiseman, Jul 13 2018

Keywords

• nonn

Comments

A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An orderless identity tree-factorization of n is either (case 1) the number n itself or (case 2) a finite set of two or more distinct orderless identity tree-factorizations, one of each factor in a factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

Examples

			The a(24)=10 orderless identity tree-factorizations:
  24
  (4*6)
  (3*8)
  (2*12)
  (2*3*4)
  (4*(2*3))
  (3*(2*4))
  (2*(2*6))
  (2*(3*4))
  (2*(2*(2*3)))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001055, A001678, A004111, A292504, A292505, A295279, A300660, A316782.

Programs

Mathematica

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltsfacs[n_]:=If[n<=1,{{}},Prepend[Select[Union@@Function[q,Sort/@Tuples[oltsfacs/@q]]/@DeleteCases[postfacs[n],{n}],UnsameQ@@#&],n]];
Table[Length[oltsfacs[n]],{n,100}]

PARI

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(w[k], e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

Formula

a(p^n) = A300660(n) for prime p. - Andrew Howroyd, Nov 18 2018