cp's OEIS Frontend

A051694 Smallest Fibonacci number that is divisible by n-th prime.

%I A051694 #43 Jul 02 2025 16:01:58
%S A051694 2,3,5,21,55,13,34,2584,46368,377,832040,4181,6765,701408733,987,
%T A051694 196418,591286729879,610,72723460248141,190392490709135,24157817,
%U A051694 8944394323791464,160500643816367088,89,7778742049,12586269025
%N A051694 Smallest Fibonacci number that is divisible by n-th prime.
%C A051694 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 A051694 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 A051694 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 A051694 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 A051694 D. D. Wall, <a href="http://www.jstor.org/stable/2309169">Fibonacci series modulo m</a>, Amer. Math. Monthly, 67 (1960), 525-532.
%F A051694 a(n) = A000045(A001602(n)). - _Max Alekseyev_, Dec 12 2007
%F A051694 log a(n) << (n log n)^2. - _Charles R Greathouse IV_, Jul 17 2012
%e A051694 55 is first Fibonacci number that is divisible by 11, the 5th prime, so a(5) = 55.
%p A051694 F:= proc(n) option remember; `if`(n<2, n, F(n-1)+F(n-2)) end:
%p A051694 a:= proc(n) option remember; local p, k; p:=ithprime(n);
%p A051694       for k while irem(F(k), p)>0 do od; F(k)
%p A051694     end:
%p A051694 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 28 2015
%t A051694 f[n_] := Block[{fib = Fibonacci /@ Range[n^2]}, Reap@ For[k = 1, k <= n, k++, Sow@ SelectFirst[fib, Mod[#, Prime@ k] == 0 &]] // Flatten //
%t A051694 Rest]; f@ 26 (* _Michael De Vlieger_, Mar 28 2015, Version 10 *)
%o A051694 (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 A051694 Cf. A000045, A001602, A001605, A005478.
%K A051694 nonn,easy
%O A051694 1,1
%A A051694 _N. J. A. Sloane_
%E A051694 More terms from _Jud McCranie_
%E A051694 More terms from _James Sellers_, Dec 08 1999