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A340123
Odd composite integers m such that A004254(2*m-J(m,21)) == J(m,21) (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
3
25, 115, 125, 253, 275, 391, 425, 505, 527, 551, 575, 625, 713, 715, 775, 779, 935, 1705, 1807, 1919, 2525, 2627, 2875, 2893, 2929, 3125, 3281, 4033, 4141, 5191, 5555, 5671, 5777, 5983, 6049, 6325, 6479, 6565, 6575, 6875, 7625, 7645, 7739, 8585, 8695, 9361, 9451, 9775
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)) == J(p,D) (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)) == 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=21 and k=2, while U(m) is 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, 10000, 2], CoprimeQ[#, 21] && CompositeQ[#] &&
Divisible[ ChebyshevU[2*# - JacobiSymbol[#, 21] - 1, 5/2] - JacobiSymbol[#, 21], #] &]
CROSSREFS
Cf. A004254, A071904, A340098 (a=5, b=1, k=1).
Cf. A340122 (a=3, b=1, k=2), A340124 (a=7, b=1, k=2).
Sequence in context: A044657 A160437 A050589 * A247683 A020152 A218493
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved