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A320267 Number of balanced complete orderless tree-factorizations of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A160322 A198293 A043281 * A304327 A307428 A061704

Adjacent sequences:  A320264 A320265 A320266 * A320268 A320269 A320270

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 08 2018

STATUS

approved

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Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)