A339130 Odd composite integers m such that A003501(m-J(m,21)) == 2 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

25, 121, 169, 275, 289, 361, 527, 529, 551, 575, 841, 961, 1369, 1681, 1807, 1849, 1919, 2209, 2783, 2809, 3025, 3481, 3721, 4033, 4489, 5041, 5329, 5777, 5983, 6049, 6241, 6479, 6575, 6889, 7267, 7645, 7921, 8959, 8993, 9361, 9409, 9775

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=5 and b=1, we have D=21 and V(m) recovers A003501(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..42.

Mathematica

Select[Range[3, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[# - JacobiSymbol[#, 21], 5/2] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)

Crossrefs

Cf. A003501.

Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339131 (a=7, b=1).

Sequence in context: A036057 A083509 A343075 * A256519 A298009 A213445

Adjacent sequences: A339127 A339128 A339129 * A339131 A339132 A339133

Keyword

nonn

Author

Ovidiu Bagdasar, Nov 24 2020

Status

approved