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A340098
Odd composite integers m such that A004254(m-J(m,21)) == 0 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
6
115, 253, 391, 527, 551, 713, 715, 779, 935, 1705, 1807, 1919, 2627, 2893, 2929, 3281, 4033, 4141, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491, 17249, 17963, 18299, 18407, 20087, 20099, 21505, 22499, 24463
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=5 and b=1, we have D=21 and U(m) recovers A004254(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, 25000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 21] - 1, 5/2], #] &]
CROSSREFS
Cf. A004254, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340099 (a=7, b=1).
Sequence in context: A277806 A122562 A063361 * A208815 A334345 A255143
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved