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.

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

Robert Israel, Table of n, a(n) for n = 1..1000

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

Adjacent sequences: A339519 A339520 A339521 * A339523 A339524 A339525

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 07 2020

Status

approved