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Odd composite integers m such that A006497(m-J(m,13)) == 2*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.
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%I #9 Nov 26 2020 23:02:11

%S 9,25,49,119,121,289,361,529,649,833,841,961,1089,1189,1369,1681,1849,

%T 1881,2023,2209,2299,2809,3025,3481,3721,4187,4489,5041,5329,6241,

%U 6889,7139,7921,9409,10201,10241,10609,11449,11881,12769,12871,13833,14041,14161

%N Odd composite integers m such that A006497(m-J(m,13)) == 2*J(m,13) (mod m), where J(m,13) 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*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.

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

%C For a=3 and b=-1, we have D=13 and V(m) recovers A006497(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, 15000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 13]), 3] - 2*j, #] &] (* _Amiram Eldar_, Nov 26 2020 *)

%Y Cf. A006497.

%Y Cf. A339125 (a=1, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Nov 24 2020