A339126 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.

9, 25, 49, 119, 121, 289, 361, 529, 649, 833, 841, 961, 1089, 1189, 1369, 1681, 1849, 1881, 2023, 2209, 2299, 2809, 3025, 3481, 3721, 4187, 4489, 5041, 5329, 6241, 6889, 7139, 7921, 9409, 10201, 10241, 10609, 11449, 11881, 12769, 12871, 13833, 14041, 14161

Offset

1,1

Comments

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

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

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

For a=3 and b=-1, we have D=13 and V(m) recovers A006497(m).

References

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

D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)

Links

Table of n, a(n) for n=1..44.

Mathematica

Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 13]), 3] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

Crossrefs

Cf. A006497.

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).

Sequence in context: A318737 A246331 A141768 * A176970 A110284 A109367

Adjacent sequences: A339123 A339124 A339125 * A339127 A339128 A339129

Keyword

nonn

Author

Ovidiu Bagdasar, Nov 24 2020

Status

approved