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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.

%I A294339 #22 Dec 01 2017 03:38:37
%S A294339 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,
%T A294339 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,
%U A294339 2,101,2,6,19,203,6,26,2,19,6,26,2,237,2,6,19,19
%N 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.
%H A294339 Hans Havermann, <a href="/A294339/b294339.txt">Table of n, a(n) for n = 1..10000</a>
%F A294339 a(n) = A294338(2^n). - _R. J. Mathar_, Nov 27 2017
%e A294339 The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.
%p A294339 f:= proc(n) option remember; local F,t,s,g,a;
%p A294339   F:= ifactors(n)[2];
%p A294339   g:= igcd(op(map(t -> t[2],F)));
%p A294339   t:= 1;
%p A294339   for s in numtheory:-divisors(g) minus {1} do
%p A294339     t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
%p A294339   od;
%p A294339   t
%p A294339 end proc:
%p A294339 seq(f(2^n),n=1..100); # _Robert Israel_, Dec 01 2017
%t A294339 a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
%t A294339 Table[a[2^n],{n,100}]
%Y A294339 Cf. A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294338.
%K A294339 nonn
%O A294339 1,2
%A A294339 _Gus Wiseman_, Oct 28 2017