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URL: https://oeis.org/A339522

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A339522
Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
4
95, 115, 145, 253, 391, 527, 551, 713, 715, 779, 935, 1045, 1615, 1705, 1805, 1807, 1919, 2185, 2627, 2755, 2893, 2929, 2945, 3281, 4033, 4141, 4205, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8441, 8555, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491
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=2, while V(m) recovers A003501(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, 24(1), 9-15 (2018).
MAPLE
filter:= proc(m)
uses LinearAlgebra:-Modular;
local p, M;
if igcd(m, 21) <> 1 then return false fi;
if isprime(m) then return false fi;
p:= 2*m - numtheory:-jacobi(m, 21);
M:= Mod(m, [[0, 1], [-1, 5]], integer[8]);
(MatrixPower(m, M, p) . <2, 5>)[1] - 5 mod m = 0
end proc:
select(filter, [seq(i, i=9..20000, 2)]); # Robert Israel, Dec 15 2020
MATHEMATICA
Select[Range[3, 20000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[2*# - JacobiSymbol[#, 21], 5/2] - 5, #] &]
CROSSREFS
Cf. A003501, A071904, A339130 (a=5, b=1, k=1).
Cf. A339521 (a=3, b=1), A339523 (a=7, b=1).
Sequence in context: A181767 A331663 A046005 * A045121 A207374 A057874
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 07 2020
STATUS
approved