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A340124 Odd composite integers m such that A004187(2*m-J(m,45)) == J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

%I A340124 #10 Jan 01 2021 14:36:10
%S A340124 49,323,329,343,377,451,1081,1127,1771,1819,1891,2033,2303,2401,3653,
%T A340124 3827,4181,5671,5777,6601,6721,7471,7931,8149,8557,9691,10877,11309,
%U A340124 11663,13201,13861,13981,14701,15251,15449,16121,16807,17119,17513,17687,17711
%N A340124 Odd composite integers m such that A004187(2*m-J(m,45)) == J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.
%C A340124 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 A340124 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 A340124 Here b=1, a=7, D=45 and k=2, while U(m) is A004187(m).
%D A340124 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340124 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340124 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340124 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 A340124 Select[Range[3, 20000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&
%t A340124 Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 45] - 1, 7/2] - JacobiSymbol[#, 45],  #] &]
%Y A340124 Cf. A004187, A071904, A340099 (a=7, b=1, k=1).
%Y A340124 Cf. A340122 (a=3, b=1, k=2), A340123 (a=5, b=1, k=2).
%K A340124 nonn
%O A340124 1,1
%A A340124 _Ovidiu Bagdasar_, Dec 28 2020