login
A339729
Odd composite integers m such that A003501(3*m-J(m,21)) == 23 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
3
25, 55, 85, 115, 155, 187, 253, 275, 341, 407, 527, 551, 559, 575, 851, 925, 1199, 1265, 1633, 1775, 1807, 1919, 1961, 2123, 2507, 2635, 2641, 2725, 3401, 3553, 3959, 4033, 4381, 4807, 5461, 5777, 5797, 5977, 5983, 6049, 6325, 6439, 6479, 6575, 7645, 7999, 8639
OFFSET
1,1
COMMENTS
The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1) (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 V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=5, D=21 and k=3, while V(m) recovers A003501(m), with V(2)=23.
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, 24(1), 9-15 (2018).
MATHEMATICA
Select[Range[3, 9000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[3*# - JacobiSymbol[#, 21], 5/2] - 23, #] &]
CROSSREFS
Cf. A003501, A071904, A339130 (a=5, b=1, k=1), A339522 (a=5, b=1, k=2).
Cf. A339728 (a=3, b=1), A339730 (a=7, b=1).
Sequence in context: A176275 A108166 A080863 * A091214 A338009 A036305
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 14 2020
STATUS
approved