A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

323, 377, 1001, 1183, 1729, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 8841, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 26011, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 41041

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 V(k*p-J(p,D)) == V(k-1)*J(p,D) (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 V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=1, D=5 and k=2, while V(m) recovers A000032(m) (Lucas numbers).

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..33.

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

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

Mathematica

Select[Range[3, 45000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 5]] - JacobiSymbol[#, 5], #] &]

Crossrefs

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1).

Cf. A339518 (a=3, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Sequence in context: A082948 A182554 A340118 * A217120 A081264 A069107

Adjacent sequences: A339514 A339515 A339516 * A339518 A339519 A339520

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 07 2020

Status

approved