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A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.

%I A340237 #8 Jan 04 2021 06:29:59
%S A340237 9,27,33,35,65,81,99,121,221,243,297,363,513,585,627,705,729,891,1089,
%T A340237 1539,1541,1881,2145,2187,2299,2673,3267,3605,4181,4573,4579,5265,
%U A340237 5633,6721,6993,7865,8019,8979,9131,9801,10307,10877,10881,13333,13741,14001,14705,14989
%N A340237 Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.
%C A340237 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 A340237 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 A340237 Here b=-1, a=5, D=29 and k=3, while U(m) is A052918(m).
%D A340237 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340237 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340237 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340237 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 A340237 Select[Range[3, 15000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 29], 5] - 5, #] &]
%Y A340237 Cf. A052918, A071904, A340095 (a=5, b=-1, k=1), A340120 (a=5, b=-1, k=2).
%Y A340237 Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340238 (a=7, b=-1, k=3).
%K A340237 nonn
%O A340237 1,1
%A A340237 _Ovidiu Bagdasar_, Jan 01 2021