cp's OEIS Frontend

A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

%I A340119 #10 Jan 01 2021 14:34:55
%S A340119 9,27,63,81,99,119,153,243,567,649,729,759,891,903,1071,1189,1377,
%T A340119 1431,1539,1763,1881,1953,2133,2187,3599,3897,4187,4585,5103,5313,
%U A340119 5559,5589,5819,6561,6681,6831,6993,8019,8127,8829,8855,9639,9999,10611,11135,11691,11961
%N A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.
%C A340119 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. 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=3, D=13 and k=2, while U(m) is A006190(m).
%D A340119 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340119 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340119 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340119 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 A340119 Select[Range[3, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]
%Y A340119 Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).
%Y A340119 Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).
%K A340119 nonn
%O A340119 1,1
%A A340119 _Ovidiu Bagdasar_, Dec 28 2020