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A340235 Odd composite integers m such that A000045(3*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol. 3
9, 27, 161, 341, 901, 1107, 1281, 1853, 2241, 2529, 4181, 5473, 5611, 5777, 6119, 6721, 7587, 8307, 9729, 10877, 11041, 12209, 13201, 13277, 14981, 15251, 16771, 17567, 20591, 20769, 20801, 22827, 23323, 24921, 27403, 28421, 29489, 33001, 34561, 38529, 38801 (list; graph; refs; listen; history; text; internal format)
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=1, D=5 and k=3, 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, 40000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 5], 1] - 1, #] &]
CROSSREFS
Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A340118 (a=1, b=-1, k=2).
Cf. A340236 (a=3, b=-1, k=3), A340237 (a=5, b=-1, k=3), A340238 (a=7, b=-1, k=3).
Sequence in context: A328604 A357667 A230185 * A217640 A227474 A286524
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jan 01 2021
STATUS
approved

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Last modified March 28 04:05 EDT 2024. Contains 371235 sequences. (Running on oeis4.)