A340119 - OEIS

A340119

Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

9, 27, 63, 81, 99, 119, 153, 243, 567, 649, 729, 759, 891, 903, 1071, 1189, 1377, 1431, 1539, 1763, 1881, 1953, 2133, 2187, 3599, 3897, 4187, 4585, 5103, 5313, 5559, 5589, 5819, 6561, 6681, 6831, 6993, 8019, 8127, 8829, 8855, 9639, 9999, 10611, 11135, 11691, 11961

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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(2*p-J(p,D)) == 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 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=2, 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..47.

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, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]

CROSSREFS

Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Sequence in context: A112524 A254622 A374976 * A271990 A153237 A256327

Adjacent sequences: A340116 A340117 A340118 * A340120 A340121 A340122

KEYWORD

nonn

AUTHOR

Ovidiu Bagdasar, Dec 28 2020

STATUS

approved