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A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

%I A339523 #17 Dec 15 2020 16:24:22
%S A339523 91,203,323,329,377,451,1001,1081,1183,1547,1729,1771,1819,1891,1967,
%T A339523 2033,2093,2639,2821,3197,3311,3653,3731,3827,4181,4669,5551,5671,
%U A339523 5777,5887,6601,6721,7471,7931,7973,8149,8557,9541,9737,10877,11309,11663,11977,13201
%N A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.
%C A339523 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 A339523 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 A339523 Here b=1, a=7, D=45 and k=2, while V(m) recovers A056854(m).
%D A339523 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339523 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A339523 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A339523 Robert Israel, <a href="/A339523/b339523.txt">Table of n, a(n) for n = 1..1000</a>
%H A339523 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).
%p A339523 filter:= proc(m)
%p A339523 uses LinearAlgebra:-Modular;
%p A339523 local p,M;
%p A339523   if igcd(m,45) <> 1 then return false fi;
%p A339523   if isprime(m) then return false fi;
%p A339523   p:= 2*m - numtheory:-jacobi(m,45);
%p A339523   M:= Mod(m,[[0,1],[-1,7]],integer[8]);
%p A339523   (MatrixPower(m,M,p) . <2,7>)[1] - 7 mod m = 0
%p A339523 end proc:
%p A339523 select(filter, [seq(i,i=9..10000,2)]); # _Robert Israel_, Dec 15 2020
%t A339523 Select[Range[3, 15000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(2*# - JacobiSymbol[#, 45])] - 7, #] &]
%Y A339523 Cf. A056854, A071904, A339131 (a=7, b=1, k=1).
%Y A339523 Cf. A339521 (a=3, b=1), A339522 (a=5, b=1).
%K A339523 nonn
%O A339523 1,1
%A A339523 _Ovidiu Bagdasar_, Dec 07 2020