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 A316782 Number of achiral tree-factorizations of n. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..10000 FORMULA a(n) = 1 + Sum_{n = d^k, k>1} a(d). a(p^n) = A067824(n) for prime p. - Andrew Howroyd, Nov 18 2018 EXAMPLE The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)). MATHEMATICA a[n_]:=1+Sum[a[d], {d, n^(1/Rest[Divisors[GCD@@FactorInteger[n][[All, 2]]]])}]; Array[a, 100] PROG (PARI) a(n)={my(z, e=ispower(n, , &z)); 1 + if(e, sumdiv(e, d, if(d

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Last modified April 4 01:40 EDT 2020. Contains 333212 sequences. (Running on oeis4.)