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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.
4
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
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
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 07 2020
STATUS
approved