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A316782
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Number of achiral tree-factorizations of n.
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13
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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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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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.
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.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018
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LINKS
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FORMULA
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a(n) = 1 + Sum_{n = d^k, k>1} a(d).
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EXAMPLE
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The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)).
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MATHEMATICA
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a[n_]:=1+Sum[a[d], {d, n^(1/Rest[Divisors[GCD@@FactorInteger[n][[All, 2]]]])}];
Array[a, 100]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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