A228439 Numbers k dividing u(k), where the Lucas sequence is defined u(i) = u(i-1) - 2*u(i-2) with initial conditions u(0)=0, u(1)=1.

1, 7, 49, 343, 2401, 4753, 16807, 33271, 76783, 117649, 232897, 461041, 537481, 823543, 1630279, 3227287, 3762367, 5764801, 7447951, 11411953, 11527201, 19358969, 22591009, 26336569, 40353607, 44720977, 52135657, 79883671, 80690407

Offset

1,2

Comments

Since the absolute value of the discriminant of the characteristic polynomial is prime (=7), the sequence contains every nonnegative integer power of 7. Other terms are formed on multiplication of 7^k by sporadic primes.

Links

Table of n, a(n) for n=1..29.

Chris Smyth, The Terms in Lucas Sequences Divisible by their Indices, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4.

Wikipedia, Lucas sequence.

Example

For k = 0, 1 , ..., 10, there is u(k) = 0,1,1,-1,-3,-1,5,7,-3,-17,-11. Clearly only k = 1 and k = 7 satisfy k divides u(k).

Mathematica

nn = 10000; s = LinearRecurrence[{1, -2}, {1, 1}, nn]; t = {}; Do[If[Mod[s[[n]], n] == 0, AppendTo[t, n]], {n, nn}]; t (* T. D. Noe, Nov 08 2013 *)

Crossrefs

Cf. A107920 (Lucas Sequence u(n)=u(n-1)-2u(n-2)).

Sequence in context: A269654 A250359 A045584 * A216130 A124536 A045578

Adjacent sequences: A228436 A228437 A228438 * A228440 A228441 A228442

Keyword

nonn,more

Author

Thomas M. Bridge, Nov 02 2013

Extensions

a(19)-a(29) from Amiram Eldar, May 28 2024

Status

approved