A339728 Odd composite integers m such that A005248(3*m-J(m,5)) == 7 (mod m), where J(m,5) is the Jacobi symbol.

9, 21, 27, 63, 161, 189, 207, 261, 287, 323, 341, 377, 671, 783, 861, 901, 987, 1007, 1107, 1269, 1281, 1287, 1449, 1853, 1891, 2071, 2241, 2407, 2431, 2501, 2529, 2567, 2743, 2961, 3201, 3827, 4181, 4623, 5029, 5473, 5611, 5777, 5781, 6119, 6601, 6721, 7161

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) (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) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.

Here b=1, a=3, D=5 and k=3, while V(m) recovers A005248(m), with V(2)=7.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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, 7500, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*(3*# - JacobiSymbol[#, 5])] - 7, #] &]

Crossrefs

Cf. A005248, A071904, A339129 (a=3, b=1, k=1), A339521 (a=3, b=1, k=2).

Cf. A339729 (a=5, b=1), A339730 (a=7, b=1).

Sequence in context: A348839 A036316 A340122 * A267216 A039289 A045252

Adjacent sequences: A339725 A339726 A339727 * A339729 A339730 A339731

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 14 2020

Status

approved