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A340121 Odd composite integers m such that A054413(2*m-J(m,53)) == 1 (mod m), where J(m,53) is the Jacobi symbol.

%I A340121 #8 Jan 01 2021 14:35:23
%S A340121 25,35,39,49,51,65,91,147,175,245,301,325,343,391,455,507,575,605,637,
%T A340121 663,741,833,897,903,935,1127,1205,1225,1247,1295,1505,1595,1633,1715,
%U A340121 1763,1775,1911,1921,2107,2275,2401,2407,2499,2599,2651,3025,3143,3185,3311
%N A340121 Odd composite integers m such that A054413(2*m-J(m,53)) == 1 (mod m), where J(m,53) is the Jacobi symbol.
%C A340121 The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
%C A340121 The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=7, D=53 and k=2, while U(m) is A054413(m).
%D A340121 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340121 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340121 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340121 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 A340121 Select[Range[3, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 53], 7] - 1, #] &]
%Y A340121 Cf. A054413, A071904, A340096 (a=7, b=-1, k=1).
%Y A340121 Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2).
%K A340121 nonn
%O A340121 1,1
%A A340121 _Ovidiu Bagdasar_, Dec 28 2020