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A337231
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Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020)
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)
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LINKS
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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.
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MATHEMATICA
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Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] &]
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CROSSREFS
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Cf. A000045.
Sequence in context: A088289 A046009 A350367 * A117223 A160355 A211712
Adjacent sequences: A337228 A337229 A337230 * A337232 A337233 A337234
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KEYWORD
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nonn
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AUTHOR
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Ovidiu Bagdasar, Aug 20 2020
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STATUS
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approved
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