A094411 -id:A094411 - cp's OEIS frontend

A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611

Offset: 1

Max Alekseyev, May 16 2012

nonn

Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2 - x - 1 is 5777". Crandall and Pomerance state that the first such Frobenius pseudoprime is actually 4181.
The Frobenius (1,-1) pseudoprimes are a subset of the odd Fibonacci pseudoprimes A081264. Among other ways, this can be seen by Theorem 3.6.6 of Crandall and Pomerance (2005) where the Frobenius criterion with respect to x^2 - Px + Q is an additional condition on an input which has passed the Lucas test for the same polynomial. - Dana Jacobsen, Aug 05 2015
Many other quadratics have a sparser set of pseudoprimes.  For example, while there are 98702 pseudoprimes below 10^13 with respect to the Fibonacci polynomial, there are only 3897 for x^2 - 3x - 5. - Dana Jacobsen, Aug 05 2015
This is the intersection of A049062 and (A081264 union A141137), that is, composite k coprime to 5 such that Fibonacci(k) == (k/5) (mod k) and that k divides Fibonacci(k-(k/5)), where (k/5) is the Legendre or Jacobi symbol. - Jianing Song, Sep 12 2018

R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..653 from Max Alekseyev)
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 89.
Jon Grantham, Frobenius pseudoprimes, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms through 10^13)
A. Rotkiewicz, Lucas and Frobenius Pseudoprimes, Annales Mathematicae Silesiane, 17 (2003): 17-39.
Lawrence Somer, Lucas sequences {Uk} for which U2p and Up are pseudoprimes for almost all primes p, Fibonacci Quart. 44 (2006), no. 1, 7-12.
Eric W. Weisstein, Frobenius Pseudoprime, MathWorld.

Cf. A049062, A081264, A141137.
Cf. also A005845, A094063, A094395, A094411.
Terms congruent to 2 or 3 mod 5 are given in A212423.

{ isFP(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x)||(kronecker(5,n)==1 && t==x) }

use ntheory ":all"; foroddcomposites { say if is_frobenius_pseudoprime($,1,-1) } 1e10; # _Dana Jacobsen, Aug 05 2015

A094412 Numbers k that divide Fibonacci(k+1) but do not divide Fibonacci(k) + 1.

Original entry on oeis.org

323, 377, 2834, 3827, 6479, 11663, 18407, 19043, 20999, 23407, 25877, 27323, 34943, 35207, 39203, 44099, 47519, 50183, 51983, 53663, 60377, 65471, 78089, 79547, 81719, 82983, 84279, 84419, 86063, 90287, 94667, 104663, 109871, 121103, 121393

Offset: 1

Eric Rowland, May 01 2004

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000045, A069107, A094395, A094401, A094411.

Select[Range[50000], ! Mod[Fibonacci[ # ] + 1, # ] == 0 && Mod[Fibonacci[ # + 1], # ] == 0 &]
Select[Range[122000],Divisible[{Fibonacci[#+1],Fibonacci[#]+1},#]=={True,False}&] (* Harvey P. Dale, Apr 16 2019 *)

fibmod(n,m)=(Mod([0,1;1,1],m)^n*[0;1])[1,1]
is(n)=fibmod(n+1,n)==0 && fibmod(n,n)!=-1 \\ Charles R Greathouse IV, Nov 04 2016

A212423 Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

Original entry on oeis.org

5777, 10877, 75077, 100127, 113573, 161027, 162133, 231703, 430127, 635627, 851927, 1033997, 1106327, 1256293, 1388903, 1697183, 2263127, 2435423, 2662277, 3175883, 3399527, 3452147, 3774377, 3900797, 4109363, 4226777, 4403027, 4828277, 4870847

Offset: 1

Max Alekseyev, May 16 2012

nonn

Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2 - x - 1 is 5777". However n = 5777 is the first Frobenius pseudoprime with respect to x^2 - x - 1 that has Jacobi symbol (5/n) = -1, i.e., n == 2,3 (mod 5). Unrestricted version with the first term 4181 is given in A212424.
Intersection of A212424 and A047221.
Composite k == 2,3 (mod 5) such that Fibonacci(k) == -1 (mod k) and that k divides Fibonacci(k+1). - Jianing Song, Sep 12 2018

R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.

Amiram Eldar, Table of n, a(n) for n = 1..14070 (terms below 10^13 from Dana Jacobsen's site)
Jon Grantham, Frobenius Pseudoprimes, Mathematics of Computation, 7 (2000), 873-891.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data.
Eric Weisstein's World of Mathematics, Frobenius Pseudoprime.

Cf. A047221, A094063, A094395, A094411, A212424.

{ isFP23(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x) }

A094413 Odd composites k that divide Fibonacci(k) + 1 but not Fibonacci(k+1).

Original entry on oeis.org

17261, 120581, 163059, 300847, 1842581, 3489641, 3792541, 5573499, 21159271, 36561959, 40779991, 63055651, 67306519, 69503037, 72125899, 129487163, 136021717, 174008141, 175094767, 247600477, 418623283, 529281911, 814454381, 902315699

Offset: 1

Eric Rowland, May 07 2004

nonn

David Radcliffe, Table of n, a(n) for n = 1..33 (first 24 terms from G. Resta)

Cf. A094395, A094411, A094412.

isok(n) = (n % 2) && ! ((fibonacci(n) + 1) % n) && (fibonacci(n+1) % n); \\ Michel Marcus, Aug 27 2013

4 more terms from Ryan Propper, Aug 03 2005

a(9)-a(24) from Giovanni Resta, Jul 20 2013

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A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

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A094412 Numbers k that divide Fibonacci(k+1) but do not divide Fibonacci(k) + 1.

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A212423 Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

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A094413 Odd composites k that divide Fibonacci(k) + 1 but not Fibonacci(k+1).

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