|
| |
|
|
A023172
|
|
Numbers n such that n divides Fibonacci(n).
|
|
22
| |
|
|
1, 5, 12, 24, 25, 36, 48, 60, 72, 96, 108, 120, 125, 144, 168, 180, 192, 216, 240, 288, 300, 324, 336, 360, 384, 432, 480, 504, 540, 552, 576, 600, 612, 625, 648, 660, 672, 684, 720, 768, 840, 864, 900, 960, 972, 1008, 1080, 1104, 1152, 1176, 1200, 1224, 1296, 1320
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Sequence contains all powers of 5, infinitely many multiples of 12 and other numbers (including some factors of Fib(5^k), e.g. 75025).
If m is in this sequence then 5*m is (since 5*m divides 5*F(m) which in turn divides F(5*m)). Also, if m is in this sequence then F(m) is in this sequence (since if gcd(F(m),m)=m then gcd(F(F(m)),F(m)) = F(gcd(F(m),m)) = F(m)). [From Max Alekseyev (maxale(AT)gmail.com), Sep 20 2009]
Comment from Max Alekseyev, Nov 29 2010: (Start)
Every term greater than 1 is a multiple of 5 or 12.
Proof. Let n>1 divide Fibonacci number F(n) and let p be the smallest prime divisor of n.
If p=2, then 3|n implying further that 4|n. Hence, 12|n.
If p=5, then 5|n.
If p is different from 2 and 5, then p divides either F(p+1) or F(p-1)
and thus p divides either F(gcd(n,p+1)) or F(gcd(n,p-1)). Minimality
of p implies that gcd(n,p-1)=1 and gcd(n,p+1)=1 (notice that p+1 being
prime implies p=2 which is not the case). Therefore, p divides F(1)=1,
a contradiction to the existence of such p. (End)
|
|
|
REFERENCES
| S. Wolfram, "A new kind of science", p. 891
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=1..500
F. Lengyel, Divisibility Properties by Multisection
|
|
|
MATHEMATICA
| a=0; b=1; c=1; Do[a=b; b=c; c=a+b; If[Mod[c, n]==0, Print[n]], {n, 3, 1500}]
Select[Range[1350], Mod[Fibonacci[ # ], # ]==0&] (from Harvey Dale)
|
|
|
PROG
| (Haskell)
import Data.List (elemIndices)
a023172 n = a023172_list !! (n-1)
a023172_list =
map (+ 1) $ elemIndices 0 $ zipWith mod (tail a000045_list) [1..]
-- Reinhard Zumkeller, Oct 13 2011
|
|
|
CROSSREFS
| Cf. A000350. See A127787 for an essentially identical sequence.
Cf. A000045, A069104, A123976, A159051.
Sequence in context: A000327 A130624 A066869 * A100479 A018806 A191831
Adjacent sequences: A023169 A023170 A023171 * A023173 A023174 A023175
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
|
|
|
EXTENSIONS
| Edited by Don Reble (djr(AT)nk.ca), Sep 07 2003
|
| |
|
|
