A178762 -id:A178762 - cp's OEIS frontend

A005478 Prime Fibonacci numbers.

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241

Offset: 1

N. J. A. Sloane

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). - N. J. A. 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
The six known safe primes 2p + 1 such that p is a Fibonacci prime are in A263880; the values of p are in A155011. There are only two known Fibonacci primes p for which 2p - 1 is also prime, namely, p = 2 and 3. Is there a reason for this bias toward prime 2p + 1 over 2p - 1 among Fibonacci primes p? - Jonathan Sondow, Nov 04 2015

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..23
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
Chris Caldwell's The Top Twenty, Fibonacci Number.
Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
Ron 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

Cf. A001605, A000045, A030426, A075736, A099000, A263880.
Subsequence of A178762.
Column k=1 of A303216.

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

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

from itertools import islice
from sympy import isprime
def A005478_gen(): # generator of terms
    a, b = 1, 1
    while True:
        if isprime(b):
            yield b
        a, b = b, a+b
A005478_list = list(islice(A005478_gen(),10)) # Chai Wah Wu, Jun 25 2024

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

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

Sequence corrected by Enoch Haga, Feb 11 2000
One more term from Jason Earls, Jul 12 2001
Comment and proof added by Jonathan Sondow, May 24 2011

A178772 Fibonacci integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81

Offset: 1

T. D. Noe, Jun 11 2010

A Fibonacci integer is a number that can be written as the product and/or quotient of Fibonacci numbers (A000045). For example, 33 is a Fibonacci integer because Fib(10) * Fib(4) / Fib(5) = 33. Of the numbers up to 100, only 8 are not Fibonacci integers: 37, 43, 53, 59, 67, 71, 73, 74, 79, 83, 86, and 97. See A178762 for the prime numbers in this sequence.
Integers of the form A065108(n)/A065108(m) for some m and n. - Charles R Greathouse IV, Jul 18 2012
Let F(x) be the number of terms of this sequence less than or equal to x. Then exp(c*sqrt(log x) - (log x)^e) < F(x) < exp(c*sqrt(log x) + (log x)^(1/6 + e)) for any e > 0, where c is this constant. Luca, Pomerance, & Wagner conjecture that 1/6 can be replaced by 0, and note that it can be replaced by 1/8 on a strong form of the abc conjecture. - Charles R Greathouse IV, Aug 31 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
Florian Luca, Carl Pomerance, Stephan Wagner, Fibonacci Integers, J. Number Theory 131 (2011) 440-457. (Preprint)

Cf. A065108, A000045.

(* This naive program being quite slow, some terms are precomputed. *) max = Fibonacci[m = 20]; Clear[f]; Do[ f[n] = True, {n, {23, 31, 46, 62, 69, 92, 93}}]; f[] = False; Do[ f[fn = Fibonacci[n]] = True; f[fk = Fibonacci[k]] = True; If[ fn*fk < max, f[fn*fk] = True]; If[ IntegerQ[fk/fn] && fk/fk < max, f[fk/fn] = True], {n, 2, m}, {k, n, m}]; fp[] := (cnt = 0; Do[ If [f[n] && f[k], If[ n*k < max, f[n*k] = True; cnt++]; If[ IntegerQ[k/n], f[k/n] = True; cnt++]], {n, 1, max}, {k, n+1, max}]; Print[cnt, " Fibonacci integers"]; cnt); FixedPoint[fp, 0]; Reap[ Do[ If[ f[n], Sow[n]], {n, 1, 100}]][[2, 1]] (* Jean-François Alcover, Mar 01 2013 *)

A201011 Primes that are Lucas primes, or that can be written as the quotient of Lucas numbers.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 31, 41, 47, 107, 199, 211, 281, 521, 1103, 2161, 2207, 2521, 3571, 5779, 9349, 9901, 14503, 90481, 103681, 3010349, 11128427, 29134601, 54018521, 261399601, 370248451, 599786069, 6643838879, 10745088481, 10749957121, 10783342081

Offset: 1

Arkadiusz Wesolowski, Jan 08 2013

nonn

			23 is in the sequence because it is prime and Lucas(12)/(Lucas(0)*Lucas(4)) = 23.

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032, A200381, A200995, A201010, A201012. Supersequence of A005479. Subsequence of A178762.

261399601 inserted by Arkadiusz Wesolowski, Feb 05 2013

A185060 Number of Fibonacci integers in the interval [1, 10^n].

Original entry on oeis.org

10, 88, 534, 2645, 11254, 42735

Offset: 1

Arkadiusz Wesolowski, Dec 25 2012

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For each fixed epsilon > 0,
exp(C*(log(10^n))^1/2 - (log(10^n))^epsilon) < a(n) < exp(C*(log(10^n))^1/2 + (log(10^n))^(1/6+epsilon)) for x sufficiently large, where C = 2*zeta(2)*sqrt(zeta(3)/(zeta(6)*log((1 + sqrt(5))/2))) = 5.15512.... (Luca, Pomerance, Wagner (2010))
The old entry a(4) = 2681 was the result of an incorrect calculation by Luca, Pomerance and Wagner. - Arkadiusz Wesolowski, Feb 05 2013

Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci integers (Banff conference in honor of Cam Stewart, May 31, 2010 to June 4, 2010.)

Cf. A178772, A178777.

e = 4; (*lst1=the terms of A178762 that are smaller than 10^e*); lst1 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 61, 89, 107, 199, 211, 233, 281, 421, 521, 1103, 1597, 2161, 2207, 2521, 3001, 3571, 5779, 9349, 9901}; lst2 = {}; q = Times @@ Complement[Prime@Range[10^e], lst1]; Do[If[GCD[q, n] == 1, AppendTo[lst2, n]], {n, 10^e}]; Table[Length@Select[lst2, # <= 10^d &], {d, e}] (* Arkadiusz Wesolowski, Feb 05 2013 *)

a(4) corrected by T. D. Noe and Arkadiusz Wesolowski, Feb 05 2013

a(5)-a(6) from Arkadiusz Wesolowski, Feb 06 2013

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A005478 Prime Fibonacci numbers.

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A178772 Fibonacci integers.

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A201011 Primes that are Lucas primes, or that can be written as the quotient of Lucas numbers.

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A185060 Number of Fibonacci integers in the interval [1, 10^n].

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