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

Author

Ovidiu Bagdasar, Nov 24 2020

Keywords

• nonn

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)

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).

Programs

Mathematica

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