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A337234
Odd composite integers m such that A006190(m)^2 == 1 (mod m).
6
9, 33, 55, 63, 99, 119, 153, 231, 385, 399, 561, 649, 935, 981, 1023, 1071, 1179, 1189, 1199, 1441, 1595, 1763, 1881, 1953, 2001, 2065, 2255, 2289, 2465, 2703, 2751, 2849, 2871, 3519, 3599, 3655, 3927, 4059, 4081, 4187, 5015, 5151, 5559, 6061, 6119, 6215, 6273, 6431
OFFSET
1,1
COMMENTS
If p is a prime, then A006190(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=3, b=-1, U(n) recovers A006190(n) ("Bronze" Fibonacci numbers).
REFERENCES
D. Andrica and 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, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 3]*Fibonacci[#, 3] - 1, #] &]
CROSSREFS
Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337233 (a=2).
Sequence in context: A280405 A111351 A065064 * A075812 A256868 A146262
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Aug 20 2020
STATUS
approved