%I #21 Jul 21 2022 03:07:48

%S 9,15,27,45,91,121,135,143,1547,1573,1935,2015,6543,6721,8099,10403,

%T 10877,10905,13319,13741,13747,14399,14705,16109,16471,18901,19043,

%U 19109,19601,19951,20591,22753,24639,26599,26937,27593

%N Odd composite integers m such that A052918(m-J(m,29)) == 0 (mod m) and gcd(m,29)=1, where J(m,29) is the Jacobi symbol.

%C 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 the identity

%C U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.

%C This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).

%C For a=5 and b=-1, we have D=29 and U(m) recovers A052918(m).

%C If even numbers greater than 2 that are coprime to 29 are allowed, then 26, 442, 6994, ... would also be terms. - _Jianing Song_, Jan 09 2021

%D D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

%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.

%H D. Andrica and O. Bagdasar, <a href="https://doi.org/10.1007/s00009-020-01653-w">On some new arithmetic properties of the generalized Lucas sequences</a>, Mediterr. J. Math., 18, 47 (2021).

%H D. Andrica and O. Bagdasar, <a href="https://doi.org/10.3390/math9080838">On generalized pseudoprimality of level k</a>, Mathematics 2021, 9(8), 838.

%t Select[Range[3,28000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 29], 5], #] &]

%Y Cf. A052918, A071904, A081264 (a=1, b=-1), A327653 (a=3, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 28 2020

%E Coprime condition added to definition by _Georg Fischer_, Jul 20 2022