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

1,6

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

Links

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

Formula

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

Example

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)))

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}]

Prog

(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

Crossrefs

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

Sequence in context: A050334 A342084 A295279 * A284974 A293222 A305008

Adjacent sequences: A316781 A316782 A316783 * A316785 A316786 A316787

Keyword

nonn

Author

Gus Wiseman, Jul 13 2018

Status

approved