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A340097
Odd composite integers m such that A001906(m-J(m,5)) == 0 (mod m) and gcd(m,5)=1, where J(m,5) is the Jacobi symbol.
6
21, 323, 329, 377, 451, 861, 1081, 1819, 1891, 2033, 2211, 3653, 3827, 4089, 4181, 5671, 5777, 6601, 6721, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 17119, 17513, 17711, 17941, 18407, 19043, 19951, 20473, 23407, 25369, 25651, 25877, 27323, 27511
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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=3 and b=1, we have D=5 and U(m) recovers A001906(m).
REFERENCES
D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
LINKS
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18, 47 (2021).
D. Andrica and O. Bagdasar, On generalized pseudoprimality of level k, Mathematics 2021, 9(8), 838.
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, 30000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 5] - 1, 3/2], #] &]
CROSSREFS
Cf. A001906, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340098 (a=5, b=1), A340099 (a=7, b=1).
Sequence in context: A001233 A145148 A214099 * A237856 A016260 A011810
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
EXTENSIONS
Coprime condition added to definition by Georg Fischer, Jul 20 2022
STATUS
approved