OFFSET
1,4
COMMENTS
A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n. These are rooted plane trees and the ordering of branches is important. For example, {{2,2},9}, {2,{2,9}}, {{2,2},{3,3}}, {6,{2,3}}, and {{2,3},6} are distinct tree-factorizations of 36, but {9,{2,2}}, {{2,9},2}, and {{3,3},{2,2}} are not.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(p^n) = A289501(n) for prime p. - Andrew Howroyd, Nov 18 2018
EXAMPLE
The a(30)=8 tree-factorizations are 30, 2*15, 2*(3*5), 3*10, 3*(2*5), 5*6, 5*(2*3), 2*3*5.
MATHEMATICA
postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1, {{}}, Prepend[Join@@Function[q, Tuples[treefacs/@q]]/@DeleteCases[postfacs[n], {n}], n]];
Table[Length[treefacs[n]], {n, 2, 83}]
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] += w[k]^e*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2017
STATUS
approved