A016089 - OEIS

%I #27 Mar 06 2019 20:24:21

%S 1,6,18,54,162,486,1458,1926,4374,5778,13122,17334,39366,52002,118098,

%T 156006,206082,354294,468018,618246,1062882,1404054,1854738,2471058,

%U 3188646,4212162,5564214,7413174,9565938,12636486,16692642,22050774

%N Numbers n such that n divides n-th Lucas number A000032(n).

%C Note that if n divides A000032(n) and p is an odd prime divisor of A000032(n), then pn divides A000032(pn) and, furthermore, p^k*n divides A000032(p^k*n) for every integer k>=0.

%C In particular, since 6 divides A000032(6) = 2*3^2, A016089 includes all terms of the geometric progression 2*3^k for k>0 (see A099856); since 18 divides A000032(18) = 2*3^3*107, A016089 includes all terms of the form 2*107^m*3^k for k>1 and m>=0; etc.

%C Terms of A016089 starting with 18 are multiples of 18. There are no other terms of the form 18p where p is prime, except for p=3 and p=107. - _Alexander Adamchuk_, May 11 2007

%H Lars Blomberg, <a href="/A016089/b016089.txt">Table of n, a(n) for n = 1..91</a>

%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.

%H C. Smyth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Smyth/smyth2.html">The terms in Lucas Sequences divisible by their indices</a>, JIS 13 (2010) #10.2.4.

%t a = 1; b = 3; Do[c = a + b; a = b; b = c; If[Mod[c, n] == 0, Print[n]], {n, 3, 2, 10^6}]

%o (PARI) is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==0 \\ _Charles R Greathouse IV_, Nov 04 2016

%Y Cf. A000032, A000204, A025192, A008776.

%Y Cf. A099856, A072378 = numbers n such that 12n divides Fibonacci(12n), A023172 = numbers n such that n divides Fibonacci(n).

%K nonn

%O 1,2

%A _Robert G. Wilson v_

%E Extended and revised by _Max Alekseyev_, May 13 2007, May 15 2008, May 16 2008