A016089 Numbers n such that n divides n-th Lucas number A000032(n).

1, 6, 18, 54, 162, 486, 1458, 1926, 4374, 5778, 13122, 17334, 39366, 52002, 118098, 156006, 206082, 354294, 468018, 618246, 1062882, 1404054, 1854738, 2471058, 3188646, 4212162, 5564214, 7413174, 9565938, 12636486, 16692642, 22050774

Offset

1,2

Comments

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.

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.

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

Links

Lars Blomberg, Table of n, a(n) for n = 1..91

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.

C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Mathematica

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

Prog

(PARI) is(n)=(Mod([0, 1; 1, 1], n)^n*[2; 1])[1, 1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Crossrefs

Cf. A000032, A000204, A025192, A008776.

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

Sequence in context: A003208 A174313 A002933 * A099856 A292295 A183913

Adjacent sequences: A016086 A016087 A016088 * A016090 A016091 A016092

Keyword

nonn

Author

Robert G. Wilson v

Extensions

Extended and revised by Max Alekseyev, May 13 2007, May 15 2008, May 16 2008

Status

approved