A074714 Primes that divide Fibonacci number F(2^k) for some k.

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.

Links

Table of n, a(n) for n=1..25.

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

Adjacent sequences: A074711 A074712 A074713 * A074715 A074716 A074717

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