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

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

Crossrefs

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.

Sequence in context: A129089 A255201 A169594 * A322327 A124315 A101113

Adjacent sequences: A294334 A294335 A294336 * A294338 A294339 A294340

Keyword

nonn

Author

_Gus Wiseman_, Oct 28 2017

Extensions

More terms from _Antti Karttunen_, Jun 12 2018

Status

approved