OFFSET
1,1
COMMENTS
If p is a prime, then A014448(p)==4 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=4, b=-1, V(n) recovers A014448(n) (even Lucas numbers).
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..1000
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).
EXAMPLE
9 is the first odd composite integer for which A014448(9)=439204==4 (mod 9).
MAPLE
M:= <<4|1>, <1|0>>:
f:= proc(n) uses LinearAlgebra:-Modular;
local A;
A:= Mod(n, M, integer[8]);
A:= MatrixPower(n, A, n);
2*A[1, 1] - 4*A[1, 2] mod n;
end proc:
select(t -> f(t) = 4 and not isprime(t), [seq(i, i=3..10^5, 2)]); # Robert Israel, Jun 19 2020
MATHEMATICA
Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[LucasL[3#] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jun 17 2020
EXTENSIONS
More terms from Jinyuan Wang, Jun 17 2020
STATUS
approved