A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

9, 27, 161, 341, 901, 1107, 1281, 1853, 2241, 2529, 4181, 5473, 5611, 5777, 6119, 6721, 7587, 8307, 9729, 10877, 11041, 12209, 13201, 13277, 14981, 15251, 16771, 17567, 20591, 20769, 20801, 22827, 23323, 24921, 27403, 28421, 29489, 33001, 34561, 38529, 38801

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 U(3*p-J(p,D)) == a (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=1, D=5 and k=3, while U(m) is A000045(m) (Fibonacci sequence).

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.

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, 40000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 5], 1] - 1, #] &]

Crossrefs

Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A340118 (a=1, b=-1, k=2).

Cf. A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Sequence in context: A328604 A357667 A230185 * A217640 A227474 A286524

Adjacent sequences: A340232 A340233 A340234 * A340236 A340237 A340238

Keyword

nonn

Author

Ovidiu Bagdasar, Jan 01 2021

Status

approved