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%I #11 Nov 19 2018 03:11:50
%S 1,1,1,2,1,1,1,2,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,
%T 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1
%N Number of achiral tree-factorizations of n.
%C A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An achiral tree-factorization of n is either (case 1) the number n itself or (case 2) a finite constant multiset of two or more achiral tree-factorizations, one of each factor in a factorization of n.
%C a(n) is also the number of ways to write n as a left-nested power-tower ((a^b)^c)^... of positive integers greater than one. For example, the a(64) = 6 ways are 64, 8^2, 4^3, 2^6, (2^3)^2, (2^2)^3.
%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H Andrew Howroyd, <a href="/A316782/b316782.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1 + Sum_{n = d^k, k>1} a(d).
%F a(p^n) = A067824(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)).
%t a[n_]:=1+Sum[a[d],{d,n^(1/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<e, a(z^d))))} \\ _Andrew Howroyd_, Nov 18 2018
%Y Cf. A001055, A001597, A003238, A067824, A089723, A214577, A281118, A289078, A292504, A294336.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jul 13 2018
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