A340236 Odd composite integers m such that A006190(3*m-J(m,13)) == 3 (mod m), where J(m,13) is the Jacobi symbol.

9, 119, 121, 187, 327, 345, 649, 705, 1003, 1089, 1121, 1189, 1881, 2091, 2299, 3553, 4187, 5461, 5565, 5841, 6165, 6485, 7107, 7139, 7145, 7467, 7991, 8321, 8449, 11041, 12705, 12871, 13833, 14041, 16109, 16851

Offset

1,1

Comments

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a (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 U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=3, D=13 and k=3, while U(m) is A006190(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..36.

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Mathematica

Select[Range[3, 15000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 13], 3] - 3, #] &]

Crossrefs

Cf. A006190, A071904, A327653 (a=3, b=-1, k=1), A340119 (a=3, b=-1, k=2).

Cf. A340235 (a=1, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).

Sequence in context: A358387 A166823 A322928 * A087984 A197544 A234964

Adjacent sequences: A340233 A340234 A340235 * A340237 A340238 A340239

Keyword

nonn

Author

Ovidiu Bagdasar, Jan 01 2021

Status

approved