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A339727 Odd composite integers m such that A086902(3*m-J(m,53)) == 51*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

%I A339727 #8 Dec 16 2020 05:53:33
%S A339727 9,25,49,51,69,91,105,143,145,153,185,221,225,325,339,391,425,441,481,
%T A339727 637,645,705,805,833,897,925,1001,1173,1189,1207,1225,1281,1299,1365,
%U A339727 1541,1633,1653,1785,1813,1921,2325,2599,2651,2769,3133,3333,3381,3605,3825,3897
%N A339727 Odd composite integers m such that A086902(3*m-J(m,53)) == 51*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.
%C A339727 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 A339727 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 A339727 Here b=-1, a=7, D=53 and k=3, while V(m) recovers A086902(m), with V(2)=51.
%D A339727 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339727 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A339727 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A339727 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 A339727 Select[Range[3, 4000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 53], 7] - 51*JacobiSymbol[#, 53], #] &]
%Y A339727 Cf. A086902, A071904, A339128 (a=7, b=-1, k=1), A339520 (a=7, b=-1, k=2).
%Y A339727 Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339726 (a=5, b=-1).
%K A339727 nonn
%O A339727 1,1
%A A339727 _Ovidiu Bagdasar_, Dec 14 2020