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Smallest Fibonacci number that is divisible by n-th prime.
12

%I #42 Oct 19 2024 15:57:32

%S 2,3,5,21,55,13,34,2584,46368,377,832040,4181,6765,701408733,987,

%T 196418,591286729879,610,72723460248141,190392490709135,24157817,

%U 8944394323791464,160500643816367088,89,7778742049,12586269025

%N Smallest Fibonacci number that is divisible by n-th prime.

%C It is conjectured that a(n) is not divisible by prime(n)^2. See Remark on p. 528 of Wall and Conjectures in CNRS links. - _Michel Marcus_, Feb 24 2016

%H Alois P. Heinz, <a href="/A051694/b051694.txt">Table of n, a(n) for n = 1..650</a> (first 100 terms from Zak Seidov)

%H Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/Mysteres-arithmetiques-de-la-suite-de-Fibonacci.html">Mystères Arithmétiques de la Suite de Fibonacci</a>, (in French), Images des Mathématiques, CNRS, 2014.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html">Fibonacci numbers with tables of F(0)-F(500)</a>

%H D. D. Wall, <a href="http://www.jstor.org/stable/2309169">Fibonacci series modulo m</a>, Amer. Math. Monthly, 67 (1960), 525-532.

%F a(n) = A000045(A001602(n)). - _Max Alekseyev_, Dec 12 2007

%F log a(n) << (n log n)^2. - _Charles R Greathouse IV_, Jul 17 2012

%e 55 is first Fibonacci number that is divisible by 11, the 5th prime, so a(5) = 55.

%p F:= proc(n) option remember; `if`(n<2, n, F(n-1)+F(n-2)) end:

%p a:= proc(n) option remember; local p, k; p:=ithprime(n);

%p for k while irem(F(k), p)>0 do od; F(k)

%p end:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 28 2015

%t f[n_] := Block[{fib = Fibonacci /@ Range[n^2]}, Reap@ For[k = 1, k <= n, k++, Sow@ SelectFirst[fib, Mod[#, Prime@ k] == 0 &]] // Flatten //

%t Rest]; f@ 26 (* _Michael De Vlieger_, Mar 28 2015, Version 10 *)

%o (PARI) a(n)=if(n==3,5,my(p=prime(n));fordiv(p^2-1,d,if(fibonacci(d)%p==0, return(fibonacci(d))))) \\ _Charles R Greathouse IV_, Jul 17 2012

%Y Cf. A000045, A001602, A001605, A005478.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Jud McCranie_

%E More terms from _James A. Sellers_, Dec 08 1999