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A005478 Prime Fibonacci numbers.
(Formerly M0741)
53
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) == 1 mod 4 for n > 2. (Proof. Otherwise 3 < a(n) = F_k == 3 mod 4. Then k == 4 mod 6 (see A079343 and A161553) and so k is not prime. But k is prime since F_k is prime and k =/= 4 - see Caldwell.)

More generally, A190949(n) == 1 mod 4. - Sloane

With the exception of 3, every term of this sequence has a prime index in the sequence of Fibonacci numbers (A000045); e.g., 5 is the fifth Fibonacci number, 13 is the seventh Fibonacci number, 89 the eleventh, etc. - Alonso del Arte, Aug 16 2013

Note: A001605 gives those indices. - Antti Karttunen, Aug 16 2013

REFERENCES

J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..23

C. Caldwell's The Top Twenty, Fibonacci Number.

R. Knott, Mathematics of the Fibonacci Series

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Eric Weisstein's World of Mathematics, Fibonacci Prime

FORMULA

a(n) = A000045(A001605(n)). A000040 INTERSECT A000045. - R. J. Mathar, Nov 01 2007

MATHEMATICA

Select[Fibonacci[Range[400]], PrimeQ] (* Alonso del Arte, Oct 13 2011 *)

PROG

(PARI) je=[]; for(n=0, 400, if(isprime(fibonacci(n)), je=concat(je, fibonacci(n)))); je

(Sage) [i for i in fibonacci_xrange(0, 10^80) if is_prime(i)] # Bruno Berselli, Jun 26 2014

CROSSREFS

Cf. A001605, A000045, A030426, A075736.

Subsequence of A178762.

Sequence in context: A139589 A152114 A139095 * A117740 A041047 A120494

Adjacent sequences:  A005475 A005476 A005477 * A005479 A005480 A005481

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Sequence corrected by Enoch Haga, Feb 11 2000

One more term from Jason Earls (zevi_35711(AT)yahoo.com), Jul 12 2001

Comment and proof added by Jonathan Sondow, May 24 2011

STATUS

approved

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Last modified July 31 23:57 EDT 2014. Contains 245103 sequences.