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A340096 Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.

%I A340096 #14 Jan 09 2021 11:49:36
%S A340096 25,35,51,65,91,175,325,391,455,575,1247,1295,1633,1763,1775,1921,
%T A340096 2275,2407,2599,2651,3367,4199,4579,4623,5629,6441,9959,10465,10825,
%U A340096 10877,12025,13021,15155,16021,18881,19019,19039,19307,19669
%N A340096 Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.
%C A340096 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 A340096 U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.
%C A340096 This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
%C A340096 For a=7 and b=-1, we have D=53 and U(m) recovers A054413(m).
%C A340096 If even numbers greater than 2 that are coprime to 53 are allowed, then 10, 50, 370, 5050, ... would also be terms. - _Jianing Song_, Jan 09 2021
%D A340096 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340096 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340096 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340096 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 A340096 Select[Range[3,20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 53], 7], #] &]
%Y A340096 Cf. A054413, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1)
%Y A340096 Cf. A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).
%K A340096 nonn
%O A340096 1,1
%A A340096 _Ovidiu Bagdasar_, Dec 28 2020