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%I #9 Jan 01 2021 14:34:39
%S 323,377,609,1891,3081,3827,4181,5777,5887,6601,6721,8149,10877,11663,
%T 13201,13601,13981,15251,17119,17711,18407,19043,23407,25877,27323,
%U 28441,28623,30889,32509,34561,34943,35207,39203,40501
%N Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.
%C 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)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
%C 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. Here b=-1, a=1, D=5 and k=2, while U(m) is A000045(m) (Fibonacci sequence).
%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H 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 Select[Range[3, 50000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 5], 1] - 1, #] &]
%Y Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).
%Y Cf. A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Dec 28 2020
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