A339725 Odd composite integers m such that A006497(3*m-J(m,13)) == 11*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

9, 27, 119, 133, 145, 165, 205, 261, 341, 393, 649, 693, 705, 901, 945, 1121, 1173, 1189, 1353, 1431, 1485, 1881, 2133, 2805, 3201, 3605, 3745, 4187, 5173, 5461, 5841, 5945, 6165, 6213, 6485, 6943, 6993, 7107, 7991, 8321, 8449, 9669, 11041, 11781, 11961, 12861

Offset: 1

Author

Ovidiu Bagdasar, Dec 14 2020

Keywords

• nonn

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=3, while V(m) recovers A006497(m), with V(2)=11.

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

• 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).

Crossrefs

Cf. A006497, A071904, A339126 (a=3, b=-1, k=1), A339518 (a=3, b=-1, k=2).

Cf. A339724 (a=1, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

Programs

Mathematica

Select[Range[3, 13000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 13], 3] - 11*JacobiSymbol[#, 13], #] &]