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%I #11 Dec 23 2020 04:11:00
%S 49,161,287,323,329,341,377,451,671,737,901,1007,1079,1081,1127,1271,
%T 1363,1541,1819,1853,1891,1927,2033,2071,2303,2407,2431,2461,2501,
%U 2567,2743,3653,3827,4181,4843,5029,5243,5473,5611,5671,5777,6119,6593,6601,6721,6923
%N Odd composite integers m such that A056854(3*m-J(m,45)) == 47 (mod m) and gcd(m,45)=1, where J(m,45) 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) (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) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
%C Here b=1, a=7, D=45 and k=3, while V(m) recovers A056854(m), with V(2)=47.
%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 Amiram Eldar, <a href="/A339730/b339730.txt">Table of n, a(n) for n = 1..1000</a>
%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, 7000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(3*# - JacobiSymbol[#, 45])] - 47, #] &]
%Y Cf. A056854, A071904, A339131 (a=7, b=1, k=1), A339523 (a=7, b=1, k=2).
%Y Cf. A339728 (a=3, b=1), A339729 (a=5, b=1).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Dec 14 2020
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