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%I #8 Jan 04 2021 06:29:46
%S 9,49,63,141,161,207,323,341,377,441,671,901,1007,1127,1281,1449,1853,
%T 1891,2071,2303,2407,2501,2743,2961,3827,4181,4623,5473,5611,5777,
%U 6119,6593,6601,6721,7161,7567,8149,8473,8961,9729,9881
%N Odd composite integers m such that A001906(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
%C 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(3*p-J(p,D)) == a*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 U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
%C Here b=1, a=3, D=5 and k=3, while U(m) is A001906(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, 2018, 24(1), 9--15.
%t Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 5] - 1, 3/2] - 3*JacobiSymbol[#, 5], #] &]
%Y Cf. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2).
%Y Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Jan 01 2021
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