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Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.
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%I #14 Dec 15 2020 10:23:59

%S 25,35,51,65,75,91,105,175,203,325,391,455,575,645,861,1247,1275,1295,

%T 1633,1763,1775,1785,1875,1921,2275,2407,2415,2599,2625,2651,3045,

%U 3367,4199,4579,4623,5629,5835,5887,6441,6699,9959,10465,10815,10825,10877,11865,12025

%N Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) 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=7, D=53 and k=2, while V(m) recovers A086902(m).

%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, 20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 53], 7] - 7*JacobiSymbol[#, 53], #] &]

%Y Cf. A086902, A071904, A339128 (a=7, b=-1, k=1).

%Y Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339529 (a=5, b=-1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 07 2020