A294339 Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

1, 2, 2, 5, 2, 6, 2, 12, 5, 6, 2, 19, 2, 6, 6, 32, 2, 19, 2, 19, 6, 6, 2, 56, 5, 6, 12, 19, 2, 26, 2, 79, 6, 6, 6, 71, 2, 6, 6, 56, 2, 26, 2, 19, 19, 6, 2, 169, 5, 19, 6, 19, 2, 56, 6, 56, 6, 6, 2, 101, 2, 6, 19, 203, 6, 26, 2, 19, 6, 26, 2, 237, 2, 6, 19, 19

Offset: 1

Author

Gus Wiseman, Oct 28 2017

Keywords

• nonn

Examples

			The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.
		

Links

• Hans Havermann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294338.

Programs

Maple

f:= proc(n) option remember; local F,t,s,g,a;
  F:= ifactors(n)[2];
  g:= igcd(op(map(t -> t[2],F)));
  t:= 1;
  for s in numtheory:-divisors(g) minus {1} do
    t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
  od;
  t
end proc:
seq(f(2^n),n=1..100); # Robert Israel, Dec 01 2017

Mathematica

a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

Formula

a(n) = A294338(2^n). - R. J. Mathar, Nov 27 2017