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A340098 Odd composite integers m such that A004254(m-J(m,21)) == 0 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

%I A340098 #8 Dec 29 2020 02:50:51
%S A340098 115,253,391,527,551,713,715,779,935,1705,1807,1919,2627,2893,2929,
%T A340098 3281,4033,4141,5191,5671,5777,5983,6049,6479,7645,7739,8695,9361,
%U A340098 11663,11815,12121,12209,12265,14491,17249,17963,18299,18407,20087,20099,21505,22499,24463
%N A340098 Odd composite integers m such that A004254(m-J(m,21)) == 0 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
%C A340098 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 the identity
%C A340098 U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
%C A340098 This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
%C A340098 For a=5 and b=1, we have D=21 and U(m) recovers A004254(m).
%D A340098 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340098 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340098 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340098 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 A340098 Select[Range[3, 25000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 21] - 1, 5/2], #] &]
%Y A340098 Cf. A004254, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340099 (a=7, b=1).
%K A340098 nonn
%O A340098 1,1
%A A340098 _Ovidiu Bagdasar_, Dec 28 2020