

A074700


a(n) = tau(F(2^n)) where tau(x) is the number of divisors of x (A000005(x)) and F(k) the kth Fibonacci number (A000045(k)).


1



1, 2, 4, 8, 16, 64, 256, 1024, 8192, 131072, 1048576
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OFFSET

1,2


COMMENTS

Is there any pattern in this sequence? It seems also that tau(F(m^n)) is a power of 2 for any m, any n > 0.
F(2^n) = L(2)L(4)L(8)...L(2^(n1)) where L(x) is the Lucas numbers. This greatly reduces the difficulty of factoring these numbers. To find a(9) one needs the factorization of F(512); this was done long ago: F(2^9) = 3 * 7 * 47 * 127 * 1087 * 2207 * 4481 * 34303 * 119809 * 73327699969 * 186812208641 * 4698167634523379875583 * 125960894984050328038716298487435392001; hence a(9) = 2^13 = 8192. Since L(512), L(1024) are completed factored the next few terms are also known. L(2048) has 1 known factor and a C411, thus the next term is at least 2^23.  Sean A. Irvine, Jun 02 2005


LINKS



MATHEMATICA



PROG

(Magma) [NumberOfDivisors(Fibonacci(2^n)): n in [1..11]]; // Vincenzo Librandi, Jun 18 2017


CROSSREFS



KEYWORD

more,nonn,hard


AUTHOR



EXTENSIONS



STATUS

approved



