A339724 - OEIS

%I #9 Dec 16 2020 05:52:52

%S 9,21,161,341,901,1281,1853,3201,4181,5473,5611,5777,6119,6721,9729,

%T 10877,11041,12209,12441,13201,14981,15251,16771,17941,20591,20769,

%U 20801,23323,25761,27403,27661,28121,28421,29489,33001,34561,38801,39281,41159,42721

%N Odd composite integers m such that A000032(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

%C The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

%C Here b=-1, a=1, D=5 and k=3, while V(m) is A000032(m) (Lucas numbers), with V(2)=3.

%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, 24(1), 9-15 (2018).

%t Select[Range[3, 43000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 5]] - 3*JacobiSymbol[#, 5], #] &]

%Y Cf. A000032, A071904, A339125 (a=1, b=-1, k=1), A339517 (a=1, b=-1, k=2).

%Y Cf. A339725 (a=3, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 14 2020