

A316782


Number of achiral treefactorizations 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 treefactorization of n is either (case 1) the number n itself or (case 2) a finite constant multiset of two or more achiral treefactorizations, one of each factor in a factorization of n.
a(n) is also the number of ways to write n as a leftnested powertower ((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 treefactorizations 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<e, a(z^d))))} \\ Andrew Howroyd, Nov 18 2018


CROSSREFS

Cf. A001055, A001597, A003238, A067824, A089723, A214577, A281118, A289078, A292504, A294336.
Sequence in context: A089723 A305253 A294336 * A326647 A326028 A294338
Adjacent sequences: A316779 A316780 A316781 * A316783 A316784 A316785


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jul 13 2018


STATUS

approved



