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A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

%I A294337 #12 Jun 12 2018 21:15:32
%S A294337 1,2,2,4,2,4,2,6,4,4,2,7,2,4,4,10,2,7,2,7,4,4,2,10,4,4,6,7,2,8,2,12,4,
%T A294337 4,4,12,2,4,4,10,2,8,2,7,7,4,2,15,4,7,4,7,2,10,4,10,4,4,2,13,2,4,7,16,
%U A294337 4,8,2,7,4,8,2,16,2,4,7,7,4,8,2,15,10,4,2,13,4,4,4,10,2,13,4,7,4,4,4,18,2,7,7,12,2,8,2,10,8
%N A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.
%H A294337 Antti Karttunen, <a href="/A294337/b294337.txt">Table of n, a(n) for n = 1..65537</a>
%F A294337 a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - _Antti Karttunen_, Jun 12 2018
%e A294337 The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.
%t A294337 a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
%t A294337 Table[a[2^n],{n,100}]
%o A294337 (PARI)
%o A294337 A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
%o A294337 A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d)));
%o A294337 A294337(n) = sumdiv(n,d,A294336(d));
%o A294337 \\ Or alternatively, after Mathematica-code as:
%o A294337 A294337(n) = A294336(2^n); \\ _Antti Karttunen_, Jun 12 2018
%Y A294337 Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.
%K A294337 nonn
%O A294337 1,2
%A A294337 _Gus Wiseman_, Oct 28 2017
%E A294337 More terms from _Antti Karttunen_, Jun 12 2018