A335670 Odd composite integers m such that A014448(m) == 4 (mod m).

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

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

• nonn

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).

Examples

			9 is the first odd composite integer for which A014448(9)=439204==4 (mod 9).
		

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).

Crossrefs

Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335671 (a=5).

Programs

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 *)

Extensions

More terms from Jinyuan Wang, Jun 17 2020