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

Author

Ovidiu Bagdasar, Dec 07 2020

Keywords

• nonn

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).

Crossrefs

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

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

Programs

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, #] &]