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

Author

Thomas M. Bridge, Nov 02 2013

Keywords

• nonn
• more

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.

Examples

			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).
		

Links

• 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.

Crossrefs

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

Programs

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 *)

Extensions

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