A339130 - OEIS

%I #11 Dec 07 2020 18:23:17

%S 25,121,169,275,289,361,527,529,551,575,841,961,1369,1681,1807,1849,

%T 1919,2209,2783,2809,3025,3481,3721,4033,4489,5041,5329,5777,5983,

%U 6049,6241,6479,6575,6889,7267,7645,7921,8959,8993,9361,9409,9775

%N Odd composite integers m such that A003501(m-J(m,21)) == 2 (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 the identity

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

%C This sequence contains the odd composite integers with V(m-J(m,D)) == 2 (mod m).

%C For a=5 and b=1, we have D=21 and V(m) recovers A003501(m).

%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)

%t Select[Range[3, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[# - JacobiSymbol[#, 21], 5/2] - 2, #] &] (* _Amiram Eldar_, Nov 26 2020 *)

%Y Cf. A003501.

%Y Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339131 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Nov 24 2020