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URL: https://oeis.org/A340237

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A340237
Odd composite integers m such that A052918(3*m-J(m,29)) == 5 (mod m), where J(m,29) is the Jacobi symbol.
3
9, 27, 33, 35, 65, 81, 99, 121, 221, 243, 297, 363, 513, 585, 627, 705, 729, 891, 1089, 1539, 1541, 1881, 2145, 2187, 2299, 2673, 3267, 3605, 4181, 4573, 4579, 5265, 5633, 6721, 6993, 7865, 8019, 8979, 9131, 9801, 10307, 10877, 10881, 13333, 13741, 14001, 14705, 14989
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=5, D=29 and k=3, while U(m) is A052918(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
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[#, 29] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 29], 5] - 5, #] &]
CROSSREFS
Cf. A052918, A071904, A340095 (a=5, b=-1, k=1), A340120 (a=5, b=-1, k=2).
Cf. A340235 (a=1, b=-1, k=3), A340236 (a=3, b=-1, k=3), A340238 (a=7, b=-1, k=3).
Sequence in context: A276003 A255343 A108107 * A216168 A036303 A391243
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jan 01 2021
STATUS
approved