A339518 - OEIS

A339518

Odd composite integers m such that A006497(2*m-J(m,13)) == 3*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

15, 75, 105, 119, 165, 255, 375, 649, 1189, 1635, 1763, 1785, 1875, 2233, 2625, 3599, 3815, 4125, 4187, 5475, 5559, 5887, 6375, 6601, 6681, 7905, 8175, 9265, 9375, 9471, 11175, 11767, 11977, 12095, 12403, 12685, 12871, 13601, 14041, 14279, 15051, 16109, 16359

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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=3, D=13 and k=2, while V(m) recovers A006497(m).

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..43.

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, 20000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 13], 3] - 3*JacobiSymbol[#, 13], #] &]

CROSSREFS

Cf. A006497, A071904, A339126 (a=3, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Sequence in context: A022675 A214453 A317657 * A296193 A135916 A211812

Adjacent sequences: A339515 A339516 A339517 * A339519 A339520 A339521

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Dec 07 2020

STATUS

approved