A320267 Number of balanced complete orderless tree-factorizations of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1

Offset

1,16

Comments

a(1) = 1 by convention.

A rooted tree is balanced if all leaves are the same distance from the root.

An orderless tree-factorization (see A292504 for definition) is complete if all leaves are prime numbers.

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) = A120803(n) for prime p. - Andrew Howroyd, Nov 18 2018

Example

The a(96) = 5 balanced complete orderless tree-factorizations:
     (2*2*2*2*2*3)
   ((2*2)*(2*2*2*3))
   ((2*3)*(2*2*2*2))
   ((2*2*2)*(2*2*3))
  ((2*2)*(2*2)*(2*3))

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1, {{}}, Prepend[Union@@Function[q, Sort/@Tuples[oltfacs/@q]]/@DeleteCases[facs[n], {n}], n]];
Table[Length[Select[oltfacs[n], And[SameQ@@Length/@Position[#, _Integer], FreeQ[#, _Integer?(!PrimeQ[#]&)]]&]], {n, 100}]

Prog

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

Crossrefs

Cf. A001055, A048816, A050336, A281118, A292505, A119262, A120803, A292504, A320160, A320266.

Sequence in context: A368330 A043281 A365634 * A304327 A307428 A368248

Adjacent sequences: A320264 A320265 A320266 * A320268 A320269 A320270

Keyword

nonn

Author

Gus Wiseman, Oct 08 2018

Status

approved