login
A340118
Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.
4
323, 377, 609, 1891, 3081, 3827, 4181, 5777, 5887, 6601, 6721, 8149, 10877, 11663, 13201, 13601, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 27323, 28441, 28623, 30889, 32509, 34561, 34943, 35207, 39203, 40501
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=1, D=5 and k=2, while U(m) is A000045(m) (Fibonacci sequence).
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, 2018, 24(1), 9--15.
MATHEMATICA
Select[Range[3, 50000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 5], 1] - 1, #] &]
CROSSREFS
Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).
Cf. A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).
Sequence in context: A082947 A082948 A182554 * A339517 A217120 A081264
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved