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Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) is the Jacobi symbol.
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%I #14 Jan 09 2021 11:49:36

%S 25,35,51,65,91,175,325,391,455,575,1247,1295,1633,1763,1775,1921,

%T 2275,2407,2599,2651,3367,4199,4579,4623,5629,6441,9959,10465,10825,

%U 10877,12025,13021,15155,16021,18881,19019,19039,19307,19669

%N Odd composite integers m such that A054413(m-J(m,53)) == 0 (mod m), where J(m,53) 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=7 and b=-1, we have D=53 and U(m) recovers A054413(m).

%C If even numbers greater than 2 that are coprime to 53 are allowed, then 10, 50, 370, 5050, ... would also be terms. - _Jianing Song_, Jan 09 2021

%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 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[3,20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 53], 7], #] &]

%Y Cf. A054413, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1)

%Y Cf. 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