%I #17 Dec 15 2020 16:24:14
%S 95,115,145,253,391,527,551,713,715,779,935,1045,1615,1705,1805,1807,
%T 1919,2185,2627,2755,2893,2929,2945,3281,4033,4141,4205,5191,5671,
%U 5777,5983,6049,6479,7645,7739,8441,8555,8695,9361,11663,11815,12121,12209,12265,14491
%N Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) 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) (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) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
%C Here b=1, a=5, D=21 and k=2, while V(m) recovers A003501(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 Robert Israel, <a href="/A339522/b339522.txt">Table of n, a(n) for n = 1..1000</a>
%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).
%p filter:= proc(m)
%p uses LinearAlgebra:-Modular;
%p local p,M;
%p if igcd(m,21) <> 1 then return false fi;
%p if isprime(m) then return false fi;
%p p:= 2*m - numtheory:-jacobi(m,21);
%p M:= Mod(m,[[0,1],[-1,5]],integer[8]);
%p (MatrixPower(m,M,p) . <2,5>)[1] - 5 mod m = 0
%p end proc:
%p select(filter, [seq(i,i=9..20000,2)]); # _Robert Israel_, Dec 15 2020
%t Select[Range[3, 20000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[2*# - JacobiSymbol[#, 21], 5/2] - 5, #] &]
%Y Cf. A003501, A071904, A339130 (a=5, b=1, k=1).
%Y Cf. A339521 (a=3, b=1), A339523 (a=7, b=1).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Dec 07 2020