A339726 Odd composite integers m such that A087130(3*m-J(m,29)) == 27*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

9, 25, 27, 33, 35, 45, 65, 81, 99, 117, 121, 161, 175, 221, 225, 297, 325, 363, 585, 645, 705, 825, 891, 1089, 1281, 1539, 1541, 1881, 2025, 2133, 2145, 2181, 2299, 2325, 2925, 3025, 3267, 3605, 3745, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7245, 7425, 7865

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=5, D=29 and k=3, while V(m) recovers A087130(m), with V(2)=27.

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..49.

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, 8000, 2],  CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 29], 5] - 27*JacobiSymbol[#, 29], #] &]

Crossrefs

Cf. A087130, A071904, A339127 (a=5, b=-1, k=1), A339519 (a=5, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339727 (a=7, b=-1).

Sequence in context: A204918 A322999 A304035 * A155109 A268576 A053850

Adjacent sequences: A339723 A339724 A339725 * A339727 A339728 A339729

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 14 2020

Status

approved