A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

91, 203, 323, 329, 377, 451, 1001, 1081, 1183, 1547, 1729, 1771, 1819, 1891, 1967, 2033, 2093, 2639, 2821, 3197, 3311, 3653, 3731, 3827, 4181, 4669, 5551, 5671, 5777, 5887, 6601, 6721, 7471, 7931, 7973, 8149, 8557, 9541, 9737, 10877, 11309, 11663, 11977, 13201

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=7, D=45 and k=2, while V(m) recovers A056854(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, 45) <> 1 then return false fi;
  if isprime(m) then return false fi;
  p:= 2*m - numtheory:-jacobi(m, 45);
  M:= Mod(m, [[0, 1], [-1, 7]], integer[8]);
  (MatrixPower(m, M, p) . <2, 7>)[1] - 7 mod m = 0
end proc:
select(filter, [seq(i, i=9..10000, 2)]); # Robert Israel, Dec 15 2020

Mathematica

Select[Range[3, 15000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(2*# - JacobiSymbol[#, 45])] - 7, #] &]

Crossrefs

Cf. A056854, A071904, A339131 (a=7, b=1, k=1).

Cf. A339521 (a=3, b=1), A339522 (a=5, b=1).

Sequence in context: A044804 A211443 A305761 * A260064 A207077 A293648

Adjacent sequences: A339520 A339521 A339522 * A339524 A339525 A339526

Keyword

nonn

Author

Ovidiu Bagdasar, Dec 07 2020

Status

approved