A339729 - OEIS

%I #12 Dec 23 2020 04:10:53

%S 25,55,85,115,155,187,253,275,341,407,527,551,559,575,851,925,1199,

%T 1265,1633,1775,1807,1919,1961,2123,2507,2635,2641,2725,3401,3553,

%U 3959,4033,4381,4807,5461,5777,5797,5977,5983,6049,6325,6439,6479,6575,7645,7999,8639

%N Odd composite integers m such that A003501(3*m-J(m,21)) == 23 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

%C 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 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 Here b=1, a=5, D=21 and k=3, while V(m) recovers A003501(m), with V(2)=23.

%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 Amiram Eldar, <a href="/A339729/b339729.txt">Table of n, a(n) for n = 1..1000</a>

%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, 24(1), 9-15 (2018).

%t Select[Range[3, 9000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[3*# - JacobiSymbol[#, 21], 5/2] - 23, #] &]

%Y Cf. A003501, A071904, A339130 (a=5, b=1, k=1), A339522 (a=5, b=1, k=2).

%Y Cf. A339728 (a=3, b=1), A339730 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 14 2020