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Smallest positive Fibonacci number divisible by n.
6

%I #39 Nov 03 2023 00:00:43

%S 1,2,3,8,5,144,21,8,144,610,55,144,13,46368,6765,144,34,144,2584,

%T 832040,21,832040,46368,144,75025,10946,14930352,46368,377,

%U 1548008755920,832040,46368,6765,34,102334155,144,4181,2584,317811,832040,6765,46368,701408733

%N Smallest positive Fibonacci number divisible by n.

%C Corollary 1 in Riasat (2011), "every positive integer divides infinitely many Fibonacci numbers," demonstrates that this sequence is infinite. - _Alonso del Arte_, Jul 27 2013

%C Note that F(0)=0 is divisible by any n > 0, hence "positive" is added in NAME. - _Zak Seidov_, Jan 07 2016

%H Alois P. Heinz, <a href="/A047930/b047930.txt">Table of n, a(n) for n = 1..1000</a> (first 500 terms from T. D. Noe)

%H Samin Riasat, <a href="https://sriasat.files.wordpress.com/2012/12/fibonacci13.pdf">Z[phi] and the Fibonacci Sequence Modulo n</a>, Mathematical Reflections 1 (2011) pp. 1-7.

%F a(A000045(k)) = A000045(k), for k > 0. - _Altug Alkan_, Jan 07 2016

%F a(n) = n * A037943(n). - _Alois P. Heinz_, Jan 08 2017

%e Fibonacci(12) = 144 is the first Fibonacci number that is divisible by 6, so a(6) = 144.

%t With[{fibs = Fibonacci[Range[150]]}, Flatten[Table[Select[fibs, Divisible[#, n] &, 1], {n, 60}]]] (* _Harvey P. Dale_, Oct 20 2011 *)

%o (PARI) a(n)=my(k); while(fibonacci(k++)%n,); fibonacci(k) \\ _Charles R Greathouse IV_, Feb 04 2014

%Y Cf. A000045, A001177, A037943.

%K easy,nice,nonn

%O 1,2

%A _J. Lowell_

%E More terms from _Patrick De Geest_, Jun 15 1999