|
%I #14 Mar 30 2012 18:58:21
%S 3,7,47,127,1087,2207,4481,21503,34303,119809,524287,65241089,
%T 167772161,1811939329,2147483647,3758096383,16074670081,73327699969,
%U 186812208641,206158430209,2142130536449,2878401282049,5703716569087,15868293545983,274367023939583
%N Primes that divide Fibonacci number F(2^k) for some k.
%C Going out to Fibonacci(2^9) gives the additional terms 73327699969, 186812208641, 4698167634523379875583, 125960894984050328038716298487435392001. - Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 08 2005
%C 21503 is a factor of Fibonacci(2^10). 524287 is a factor of Fibonacci(2^19). 65241089 is a factor of Fibonacci(2^13). -_Donovan Johnson_, Feb 21 2008
%C From the divisibility properties of Fibonacci numbers, if a prime divides F(2^k), then it divides F(2^m) for all m >= k. The smallest value of k for these primes is 2, 3, 4, 7, 6, 5, 6, 10, 9, 8, 19, 13, 24, 23, 31, 29, 20, 9, 7, 32, 15, 16, 36, 29, 24. Every integer > 1 will occur as k because every Fibonacci other than F(0), F(1), F(6), and F(12) has a primitive prime factor.
%e F(2^5)= 3*7*47*2207 hence 3,7,47,2207 are in the sequence.
%o (PARI) forprime(p=3,10^5,if(lift((matrix(2,2,i,j,Mod(i+j<4,p))^(2^(valuation(p*p-1,2)-1)))[1,2])==0,print1(p","))) - Robert Gerbicz, Dec 17 2010
%K nonn
%O 1,1
%A _Benoit Cloitre_, Sep 04 2002
%E 3 more terms from _Donovan Johnson_, Feb 21 2008
%E a(13)-a(25) from _Robert Gerbicz_, Dec 17 2010
|
