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

2, 3, 5, 21, 55, 13, 34, 2584, 46368, 377, 832040, 4181, 6765, 701408733, 987, 196418, 591286729879, 610, 72723460248141, 190392490709135, 24157817, 8944394323791464, 160500643816367088, 89, 7778742049, 12586269025

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..650 (first 100 terms from Zak Seidov)
• Shalom Eliahou, Mystères Arithmétiques de la Suite de Fibonacci, (in French), Images des Mathématiques, CNRS, 2014.
• Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
• D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Crossrefs

Cf. A000045, A001602, A001605, A005478.

Programs

Maple

F:= proc(n) option remember; `if`(n<2, n, F(n-1)+F(n-2)) end:
a:= proc(n) option remember; local p, k; p:=ithprime(n);
      for k while irem(F(k), p)>0 do od; F(k)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 28 2015

Mathematica

f[n_] := Block[{fib = Fibonacci /@ Range[n^2]}, Reap@ For[k = 1, k <= n, k++, Sow@ SelectFirst[fib, Mod[#, Prime@ k] == 0 &]] // Flatten //
Rest]; f@ 26 (* Michael De Vlieger, Mar 28 2015, Version 10 *)

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

Formula

a(n) = A000045(A001602(n)). - Max Alekseyev, Dec 12 2007

log a(n) << (n log n)^2. - Charles R Greathouse IV, Jul 17 2012

Extensions

More terms from Jud McCranie

More terms from James Sellers, Dec 08 1999