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A340123 Odd composite integers m such that A004254(2*m-J(m,21)) == J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

%I A340123 #12 Jan 01 2021 14:36:03
%S A340123 25,115,125,253,275,391,425,505,527,551,575,625,713,715,775,779,935,
%T A340123 1705,1807,1919,2525,2627,2875,2893,2929,3125,3281,4033,4141,5191,
%U A340123 5555,5671,5777,5983,6049,6325,6479,6565,6575,6875,7625,7645,7739,8585,8695,9361,9451,9775
%N A340123 Odd composite integers m such that A004254(2*m-J(m,21)) == J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
%C A340123 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)) == J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
%C A340123 The composite integers m with the property U(k*m-J(m,D)) == J(m,D)*U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
%C A340123 Here b=1, a=5, D=21 and k=2, while U(m) is A004254(m).
%D A340123 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340123 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340123 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340123 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 A340123 Select[Range[3, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] &&
%t A340123 Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 21] - 1, 5/2] - JacobiSymbol[#, 21],  #] &]
%Y A340123 Cf. A004254, A071904, A340098 (a=5, b=1, k=1).
%Y A340123 Cf. A340122 (a=3, b=1, k=2), A340124 (a=7, b=1, k=2).
%K A340123 nonn
%O A340123 1,1
%A A340123 _Ovidiu Bagdasar_, Dec 28 2020