A339127 Odd composite integers m such that A087130(m-J(m,29)) == 2*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

9, 25, 27, 49, 81, 121, 169, 175, 225, 243, 289, 325, 361, 529, 637, 729, 961, 1225, 1331, 1369, 1539, 1681, 1849, 2025, 2209, 2809, 3025, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6721, 6889, 6929, 7921, 8281, 9409

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=5 and b=-1, we have D=29 and V(m) recovers A087130(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..41.

Mathematica

Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 29]), 5] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

Crossrefs

Cf. A087130.

Cf. A339125 (a=1, b=-1), A339126 (a=3, 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: A319165 A319152 A244623 * A117580 A280609 A340238

Adjacent sequences: A339124 A339125 A339126 * A339128 A339129 A339130

Keyword

nonn

Author

Ovidiu Bagdasar, Nov 24 2020

Status

approved