A337231 Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.

231, 323, 377, 1443, 1551, 1891, 2737, 2849, 3289, 3689, 3827, 4181, 4879, 5777, 6479, 6601, 6721, 7743, 8149, 9879, 10877, 11663, 13201, 13981, 15251, 15301, 17119, 17261, 17711, 18407, 19043, 20999, 23407, 25877, 27071, 27323, 29281, 30889, 34561, 34943, 35207

Offset

1,1

Comments

If p is a prime, then A000045(p)^2==1 (mod p).

This sequence contains the odd composite integers for which the congruence holds.

The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p)==1 (mod p) whenever p is prime and b=-1.

For a=1, b=-1, U(n) recovers A000045(n) (Fibonacci numbers).

References

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Mathematica

Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] &]

Prog

(PARI) lista(nn) = my(list=List()); forcomposite(c=1, nn, if ((c%2) && (Mod(fibonacci(c), c)^2 == 1), listput(list, c))); Vec(list); \\ Michel Marcus, Sep 29 2023

Crossrefs

Cf. A000045.

Sequence in context: A088289 A046009 A350367 * A117223 A160355 A211712

Adjacent sequences: A337228 A337229 A337230 * A337232 A337233 A337234

Keyword

nonn

Author

Ovidiu Bagdasar, Aug 20 2020

Status

approved