A340118 Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

323, 377, 609, 1891, 3081, 3827, 4181, 5777, 5887, 6601, 6721, 8149, 10877, 11663, 13201, 13601, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 27323, 28441, 28623, 30889, 32509, 34561, 34943, 35207, 39203, 40501

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(2*p-J(p,D)) == 1 (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=2, 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..34.

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, 50000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 5], 1] - 1, #] &]

Crossrefs

Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Sequence in context: A082947 A082948 A182554 * A339517 A217120 A081264

Adjacent sequences: A340115 A340116 A340117 * A340119 A340120 A340121

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 28 2020

Status

approved