OFFSET
1,6
COMMENTS
A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An orderless identity tree-factorization of n is either (case 1) the number n itself or (case 2) a finite set of two or more distinct orderless identity 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) = A300660(n) for prime p. - Andrew Howroyd, Nov 18 2018
EXAMPLE
The a(24)=10 orderless identity tree-factorizations:
24
(4*6)
(3*8)
(2*12)
(2*3*4)
(4*(2*3))
(3*(2*4))
(2*(2*6))
(2*(3*4))
(2*(2*(2*3)))
MATHEMATICA
postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
oltsfacs[n_]:=If[n<=1, {{}}, Prepend[Select[Union@@Function[q, Sort/@Tuples[oltsfacs/@q]]/@DeleteCases[postfacs[n], {n}], UnsameQ@@#&], n]];
Table[Length[oltsfacs[n]], {n, 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, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(w[k], e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 13 2018
STATUS
approved