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A340240 Odd composite integers m such that A004254(3*m-J(m,21)) == 5*J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

%I A340240 #8 Jan 04 2021 06:29:41
%S A340240 55,407,527,529,551,559,965,1199,1265,1633,1807,1919,1961,3401,3959,
%T A340240 4033,4381,5461,5777,5977,5983,6049,6233,6439,6479,7141,7195,7645,
%U A340240 7999,8639,8695,8993,9265,9361,11663,11989,12209,12265,13019,13021,13199,14023,14465,14491
%N A340240 Odd composite integers m such that A004254(3*m-J(m,21)) == 5*J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
%C A340240 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 A340240 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 A340240 Here b=1, a=5, D=21 and k=3, while U(m) is A004254(m).
%D A340240 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340240 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340240 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340240 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 A340240 Select[Range[3, 15000, 2], CoprimeQ[#, 21] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 21] - 1, 5/2] - 5*JacobiSymbol[#, 21],  #] &]
%Y A340240 Cf. A004254, A071904, A340098 (a=5, b=1, k=1), A340123 (a=5, b=1, k=2).
%Y A340240 Cf. A340239 (a=3, b=1, k=3), A340241 (a=7, b=1, k=3).
%K A340240 nonn
%O A340240 1,1
%A A340240 _Ovidiu Bagdasar_, Jan 01 2021