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A316789 Number of same-tree-factorizations of n.

%I A316789 #8 Nov 18 2018 14:47:32
%S A316789 1,1,1,2,1,1,1,2,2,1,1,1,1,1,1,6,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,
%T A316789 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,14,1,1,1,1,
%U A316789 1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1
%N A316789 Number of same-tree-factorizations of n.
%C A316789 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). A same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite sequence of two or more same-tree-factorizations, one of each factor in a constant factorization of n.
%C A316789 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H A316789 Andrew Howroyd, <a href="/A316789/b316789.txt">Table of n, a(n) for n = 1..10000</a>
%F A316789 a(n) = 1 + Sum_{n = x^y, y > 1} a(x)^y.
%F A316789 a(2^n) = A281145(n).
%e A316789 The a(64) = 14 same-tree-factorizations:
%e A316789   64
%e A316789   (8*8)
%e A316789   (4*4*4)
%e A316789   (8*(2*2*2))
%e A316789   ((2*2*2)*8)
%e A316789   (4*4*(2*2))
%e A316789   (4*(2*2)*4)
%e A316789   ((2*2)*4*4)
%e A316789   (2*2*2*2*2*2)
%e A316789   (4*(2*2)*(2*2))
%e A316789   ((2*2)*4*(2*2))
%e A316789   ((2*2)*(2*2)*4)
%e A316789   ((2*2*2)*(2*2*2))
%e A316789   ((2*2)*(2*2)*(2*2))
%t A316789 a[n_]:=1+Sum[a[n^(1/d)]^d,{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
%t A316789 Array[a,100]
%o A316789 (PARI) a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, a(z^(e/d))^d)))} \\ _Andrew Howroyd_, Nov 18 2018
%Y A316789 Cf. A001055, A001597, A001678, A003238, A007916, A052409, A052410, A067824, A089723, A281118, A281145, A294336, A316790.
%K A316789 nonn
%O A316789 1,4
%A A316789 _Gus Wiseman_, Jul 14 2018