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A320266 Number of balanced orderless tree-factorizations of n. 2
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 3, 4, 1, 5, 1, 9, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 17, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 13, 1, 2, 4, 19, 2, 5, 1, 4, 2, 5, 1, 24, 1, 2, 4, 4, 2, 5, 1, 17, 6, 2, 1, 13, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless 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) = A320160(n) for prime p. - Andrew Howroyd, Nov 18 2018

EXAMPLE

The a(36) = 11 balanced orderless tree-factorizations:

  36,

  (2*18), (3*12), (4*9), (6*6),

  (2*2*9), (2*3*6), (3*3*4),

  (2*2*3*3), ((2*2)*(3*3)), ((2*3)*(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], SameQ@@Length/@Position[#, _Integer]&]], {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, 1), v=vector(n)); while(u, v+=u; u[1]=1; u=MultEulerT(u)-u); v} \\ Andrew Howroyd, Nov 18 2018

CROSSREFS

Cf. A000669, A001055, A048816, A050336, A281118, A292505, A119262, A120803, A141268, A292504, A319312, A320160, A320267.

Sequence in context: A001055 A335079 A337093 * A277692 A129138 A112970

Adjacent sequences:  A320263 A320264 A320265 * A320267 A320268 A320269

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 08 2018

STATUS

approved

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Last modified May 6 09:25 EDT 2021. Contains 343580 sequences. (Running on oeis4.)