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%I #9 Nov 18 2018 14:57:34
%S 1,1,1,2,1,1,1,2,2,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,
%T 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1
%N Number of orderless same-tree-factorizations of n.
%C A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). An orderless same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless same-tree-factorizations, one of each factor in a constant factorization of n.
%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H Andrew Howroyd, <a href="/A316790/b316790.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1 + Sum_{n = x^y, y > 1} binomial(a(x) + y - 1, y).
%F a(2^n) = A289078(n).
%e The a(64) = 9 orderless same-tree-factorizations:
%e 64
%e (8*8)
%e (4*4*4)
%e (4*4*(2*2))
%e (8*(2*2*2))
%e (2*2*2*2*2*2)
%e (4*(2*2)*(2*2))
%e ((2*2*2)*(2*2*2))
%e ((2*2)*(2*2)*(2*2))
%t a[n_]:=1+Sum[Binomial[a[n^(1/d)]+d-1,d],{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
%t Array[a,100]
%o (PARI) a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, binomial(a(z^(e/d)) + d - 1, d))))} \\ _Andrew Howroyd_, Nov 18 2018
%Y Cf. A001055, A001597, A003238, A052409, A052410, A067824, A089723, A281118, A281145, A289078, A292504, A316789.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jul 14 2018
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