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A074714
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Primes that divide Fibonacci number F(2^k) for some k.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 (donovan.johnson(AT)yahoo.com), 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.
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EXAMPLE
| F(2^5)= 3*7*47*2207 hence 3,7,47,2207 are in the sequence.
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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
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CROSSREFS
| Sequence in context: A132565 A129518 A007670 * A064457 A005650 A020754
Adjacent sequences: A074711 A074712 A074713 * A074715 A074716 A074717
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 04 2002
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EXTENSIONS
| 3 more terms from Donovan Johnson (donovan.johnson(AT)yahoo.com), Feb 21 2008
a(13)-a(25) from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Dec 17 2010
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