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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 08 2018
STATUS
approved