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A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

%I A340235 #11 Jan 04 2021 06:29:25
%S A340235 9,27,161,341,901,1107,1281,1853,2241,2529,4181,5473,5611,5777,6119,
%T A340235 6721,7587,8307,9729,10877,11041,12209,13201,13277,14981,15251,16771,
%U A340235 17567,20591,20769,20801,22827,23323,24921,27403,28421,29489,33001,34561,38529,38801
%N A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.
%C A340235 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.
%C A340235 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 A340235 Here b=-1, a=1, D=5 and k=3, while U(m) is A000045(m) (Fibonacci sequence).
%D A340235 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340235 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340235 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340235 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 A340235 Select[Range[3, 40000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 5], 1] - 1, #] &]
%Y A340235 Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A340118 (a=1, b=-1, k=2).
%Y A340235 Cf. A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).
%K A340235 nonn
%O A340235 1,1
%A A340235 _Ovidiu Bagdasar_, Jan 01 2021