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

1,1

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

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

Sequence in context: A348316 A275394 A236385 * A196434 A197197 A295118

Adjacent sequences: A335667 A335668 A335669 * A335671 A335672 A335673

Keyword

nonn

Author

Ovidiu Bagdasar, Jun 17 2020

Extensions

More terms from Jinyuan Wang, Jun 17 2020

Status

approved