A340098 Odd composite integers m such that A004254(m-J(m,21)) == 0 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

115, 253, 391, 527, 551, 713, 715, 779, 935, 1705, 1807, 1919, 2627, 2893, 2929, 3281, 4033, 4141, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491, 17249, 17963, 18299, 18407, 20087, 20099, 21505, 22499, 24463

Offset

1,1

Comments

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy the identity

U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.

This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).

For a=5 and b=1, we have D=21 and U(m) recovers A004254(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..43.

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Mathematica

Select[Range[3, 25000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 21] - 1, 5/2], #] &]

Crossrefs

Cf. A004254, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340099 (a=7, b=1).

Sequence in context: A277806 A122562 A063361 * A208815 A334345 A255143

Adjacent sequences: A340095 A340096 A340097 * A340099 A340100 A340101

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved