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A335670
Odd composite integers m such that A014448(m) == 4 (mod m).
8
9, 85, 161, 341, 705, 897, 901, 1105, 1281, 1853, 2465, 2737, 3745, 4181, 4209, 4577, 5473, 5611, 5777, 6119, 6721, 9701, 9729, 10877, 11041, 12209, 12349, 13201, 13481, 14981, 15251, 16185, 16545, 16771, 19669, 20591, 20769, 20801, 21845, 23323, 24465, 25345
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
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
Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335671 (a=5).
Sequence in context: A348316 A275394 A236385 * A196434 A197197 A295118
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jun 17 2020
EXTENSIONS
More terms from Jinyuan Wang, Jun 17 2020
STATUS
approved