A316782 - OEIS

A316782

Number of achiral tree-factorizations of n.

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

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