%I #14 Dec 15 2020 10:24:50

%S 15,75,105,119,165,255,375,649,1189,1635,1763,1785,1875,2233,2625,

%T 3599,3815,4125,4187,5475,5559,5887,6375,6601,6681,7905,8175,9265,

%U 9375,9471,11175,11767,11977,12095,12403,12685,12871,13601,14041,14279,15051,16109,16359

%N Odd composite integers m such that A006497(2*m-J(m,13)) == 3*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

%C The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

%C The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

%C Here b=-1, a=3, D=13 and k=2, while V(m) recovers A006497(m).

%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

%D D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).

%D D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

%H Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, <a href="https://doi.org/10.1016/j.ajmsc.2017.06.002">On Fibonacci and Lucas sequences modulo a prime and primality testing</a>, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

%t Select[Range[3, 20000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 13], 3] - 3*JacobiSymbol[#, 13], #] &]

%Y Cf. A006497, A071904, A339126 (a=3, b=-1, k=1).

%Y Cf. A339517 (a=1, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 07 2020