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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.
10
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, 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, 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
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - Antti Karttunen, Jun 12 2018
EXAMPLE
The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.
MATHEMATICA
a[n_]:=1+Sum[a[g], {g, Rest[Divisors[GCD@@FactorInteger[n][[All, 2]]]]}];
Table[a[2^n], {n, 100}]
PROG
(PARI)
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n, n, sumdiv(A052409(n), d, A294336(d)));
A294337(n) = sumdiv(n, d, A294336(d));
\\ Or alternatively, after Mathematica-code as:
A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2017
EXTENSIONS
More terms from Antti Karttunen, Jun 12 2018
STATUS
approved