cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005478 Prime Fibonacci numbers.

Original entry on oeis.org

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
Offset: 1

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Author

N. J. A. Sloane

Keywords

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). - 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

References

  • 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).

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Fibonacci[Range[400]], PrimeQ] (* Alonso del Arte, Oct 13 2011 *)
  • PARI
    je=[]; for(n=0,400, if(isprime(fibonacci(n)),je=concat(je,fibonacci(n)))); je
  • Python
    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
  • Sage
    [i for i in fibonacci_xrange(0,10^80) if is_prime(i)] # Bruno Berselli, Jun 26 2014

Formula

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

Extensions

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

A263880 Safe primes 2p + 1 such that p is a Fibonacci prime.

Original entry on oeis.org

5, 7, 11, 179, 467, 21195998530602981465199287343010006825031720870818843865120019360285948694390966280586508792391539752259819
Offset: 1

Views

Author

Jonathan Sondow, Nov 02 2015

Keywords

Comments

Same as safe primes q whose Sophie Germain prime (2q - 1)/2 is a Fibonacci number.
No other terms up to 2*Fibonacci(2904353) + 1, according to the list of indices of 49 Fibonacci (probable) primes in A001605.
In that range, the only safe Fibonacci prime is 5. Are there larger ones?
There are six primes 2p + 1 such that p is a Fibonacci prime, namely, a(1) through a(6). By contrast, in the same range there are only two primes 2p - 1 such that p is a Fibonacci prime, namely, 2p - 1 = 3 and 5, for p = 2 and 3. Is there some modular restriction to explain this bias in favor of 2p + 1 over 2p - 1 among Fibonacci primes p?

Examples

			179 is in the sequence because it is prime and (179 - 1)/2 = 89 = Fibonacci(11), which is also prime.

Links

Crossrefs

Cf. A000045, A005384, A005385, A005478, A155011.

Programs

  • Mathematica
    2 * Select[Fibonacci[Range[2000]], And @@ PrimeQ[{#, 2 # + 1}] &] + 1

Formula

a(n) = 2*A155011(n) + 1.

A155012 Fibonacci prime numbers f, 3*f+2 are also primes.

Original entry on oeis.org

3, 5, 13, 89, 233, 1597, 514229, 99194853094755497, 19134702400093278081449423917
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

Keywords

Comments

3*3+2=11, 3*5+2=17, 3*13+2=41, ...

Crossrefs

Cf. A005478, A000045, A005384, A155011

Programs

  • Mathematica
    a={};Do[f=Fibonacci[n];If[PrimeQ[f],If[PrimeQ[3*f+2],AppendTo[a,f]]],{n,4*6!}];a
  • Python
    from gmpy2 import is_prime
A155012_list = []
a, b, a2, b2 = 0, 1, 2, 5
for _ in range(10**3):
    if is_prime(b) and is_prime(b2):
        A155012_list.append(b)
    a, b, a2, b2 = b, a+b, b2, a2+b2-2 # Chai Wah Wu, Nov 04 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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