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A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

%I A340236 #8 Jan 04 2021 06:30:05
%S A340236 9,119,121,187,327,345,649,705,1003,1089,1121,1189,1881,2091,2299,
%T A340236 3553,4187,5461,5565,5841,6165,6485,7107,7139,7145,7467,7991,8321,
%U A340236 8449,11041,12705,12871,13833,14041,16109,16851
%N A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.
%C A340236 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 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.
%C A340236 Here b=-1, a=3, D=13 and k=3, while U(m) is A006190(m).
%D A340236 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340236 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340236 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340236 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 A340236 Select[Range[3, 15000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 13], 3] - 3, #] &]
%Y A340236 Cf. A006190, A071904, A327653 (a=3, b=-1, k=1), A340119 (a=3, b=-1, k=2).
%Y A340236 Cf. A340235 (a=1, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).
%K A340236 nonn
%O A340236 1,1
%A A340236 _Ovidiu Bagdasar_, Jan 01 2021