login
A074714
Primes that divide Fibonacci number F(2^k) for some k.
0
3, 7, 47, 127, 1087, 2207, 4481, 21503, 34303, 119809, 524287, 65241089, 167772161, 1811939329, 2147483647, 3758096383, 16074670081, 73327699969, 186812208641, 206158430209, 2142130536449, 2878401282049, 5703716569087, 15868293545983, 274367023939583
OFFSET
1,1
COMMENTS
Going out to Fibonacci(2^9) gives the additional terms 73327699969, 186812208641, 4698167634523379875583, 125960894984050328038716298487435392001. - Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 08 2005
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
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.
EXAMPLE
F(2^5)= 3*7*47*2207 hence 3,7,47,2207 are in the sequence.
PROG
(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
CROSSREFS
Sequence in context: A129518 A007670 A263806 * A064457 A318087 A005650
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Sep 04 2002
EXTENSIONS
3 more terms from Donovan Johnson, Feb 21 2008
a(13)-a(25) from Robert Gerbicz, Dec 17 2010
STATUS
approved