A339129 Odd composite integers m such that A005248(m-J(m,5)) == 2 (mod m), where J(m,5) is the Jacobi symbol.

9, 49, 63, 121, 169, 289, 323, 361, 377, 441, 529, 841, 961, 1127, 1369, 1681, 1849, 1891, 2209, 2303, 2809, 2961, 3481, 3721, 3751, 3827, 4181, 4489, 4901, 4961, 5041, 5329, 5491, 5777, 6137, 6241, 6601, 6721, 6889, 7381, 7921, 8149, 9409, 10201, 10609, 10877, 10933, 11449, 11663

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

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

For a=3 and b=1, we have D=5 and V(m) recovers A005248(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..49.

Mathematica

Select[Range[3, 12000, 2], CompositeQ[#] && Divisible[LucasL[2*(# - JacobiSymbol[#, 5])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)

Crossrefs

Cf. A005248.

Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Sequence in context: A012260 A262537 A133478 * A340239 A032589 A137175

Adjacent sequences: A339126 A339127 A339128 * A339130 A339131 A339132

Keyword

nonn

Author

Ovidiu Bagdasar, Nov 24 2020

Status

approved