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A337231 Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number. 11
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Aug 20 2020
STATUS
approved

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Last modified July 24 09:00 EDT 2024. Contains 374575 sequences. (Running on oeis4.)