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%I #25 May 28 2024 18:29:14
%S 1,7,49,343,2401,4753,16807,33271,76783,117649,232897,461041,537481,
%T 823543,1630279,3227287,3762367,5764801,7447951,11411953,11527201,
%U 19358969,22591009,26336569,40353607,44720977,52135657,79883671,80690407
%N 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.
%C 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.
%H Chris Smyth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Smyth/smyth2.html">The Terms in Lucas Sequences Divisible by their Indices</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>.
%e 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).
%t 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 *)
%Y Cf. A107920 (Lucas Sequence u(n)=u(n-1)-2u(n-2)).
%K nonn,more
%O 1,2
%A _Thomas M. Bridge_, Nov 02 2013
%E a(19)-a(29) from _Amiram Eldar_, May 28 2024