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A340119
Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.
4
9, 27, 63, 81, 99, 119, 153, 243, 567, 649, 729, 759, 891, 903, 1071, 1189, 1377, 1431, 1539, 1763, 1881, 1953, 2133, 2187, 3599, 3897, 4187, 4585, 5103, 5313, 5559, 5589, 5819, 6561, 6681, 6831, 6993, 8019, 8127, 8829, 8855, 9639, 9999, 10611, 11135, 11691, 11961
OFFSET
1,1
COMMENTS
The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=3, D=13 and k=2, while U(m) is A006190(m).
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
LINKS
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.
MATHEMATICA
Select[Range[3, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]
CROSSREFS
Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).
Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).
Sequence in context: A112524 A254622 A374976 * A271990 A153237 A256327
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved