OFFSET
2,29
COMMENTS
A strict tree-factorization (see A295279 for definition) is complete if its leaves are all prime numbers.
From Andrew Howroyd, Nov 18 2018: (Start)
a(n) depends only on the prime signature of n.
This sequence is very similar but not identical to the number of complete orderless identity tree-factorizations of n. The first difference is at n=900 (square of three primes). Here a(n) = 191 whereas the other sequence would have 197. (End)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..10000
FORMULA
EXAMPLE
The a(72) = 6 complete strict tree-factorizations are: 2*3*(2*(2*3)), 2*(2*3*(2*3)), 2*(2*(3*(2*3))), 2*(3*(2*(2*3))), 3*(2*(2*(2*3))), (2*3)*(2*(2*3)).
MATHEMATICA
postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
sftc[n_]:=Prepend[Join@@Function[fac, Tuples[sftc/@fac]]/@Select[postfacs[n], And[Length[#]>1, UnsameQ@@#]&], n];
Table[Length[Select[sftc[n], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {n, 2, 100}]
PROG
(PARI) seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w[2..n]} \\ Andrew Howroyd, Nov 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2017
STATUS
approved