OFFSET
1,6
COMMENTS
A strict tree-factorization of n is either (case 1) the number n itself or (case 2) a set of two or more strict tree-factorizations, one of each factor in a factorization of n into distinct factors greater than one.
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
EXAMPLE
The a(30) = 8 strict tree-factorizations are: 30, (2*3*5), (2*15), (2*(3*5)), (3*10), (3*(2*5)), (5*6), (5*(2*3)).
The a(36) = 12 strict tree-factorizations are: 36, (2*3*6), (2*3*(2*3)), (2*18), (2*(2*9)), (2*(3*6)), (2*(3*(2*3))), (3*12), (3*(2*6)), (3*(2*(2*3))), (3*(3*4)), (4*9).
MATHEMATICA
sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
sft[n_]:=1+Total[Function[fac, Times@@sft/@fac]/@Select[sfs[n], Length[#]>1&]];
Array[sft, 100]
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, v[j]+=w[k]*v[j/k])); w} \\ Andrew Howroyd, Nov 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2017
STATUS
approved