A339724 Odd composite integers m such that A000032(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

9, 21, 161, 341, 901, 1281, 1853, 3201, 4181, 5473, 5611, 5777, 6119, 6721, 9729, 10877, 11041, 12209, 12441, 13201, 14981, 15251, 16771, 17941, 20591, 20769, 20801, 23323, 25761, 27403, 27661, 28121, 28421, 29489, 33001, 34561, 38801, 39281, 41159, 42721

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=3, while V(m) is A000032(m) (Lucas numbers), with V(2)=3.

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..40.

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, 43000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 5]] - 3*JacobiSymbol[#, 5], #] &]

Crossrefs

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1), A339517 (a=1, b=-1, k=2).

Cf. A339725 (a=3, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

Sequence in context: A230648 A177125 A050860 * A342409 A329007 A368120

Adjacent sequences: A339721 A339722 A339723 * A339725 A339726 A339727

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 14 2020

Status

approved