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A075702 n-th prime divides the n-th Fibonacci number. 4
2160, 3048, 27094, 251712, 505768, 936240, 2182656, 2582372, 487568736, 1261336587, 1424530096 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(12) > 2*10^9. - Giovanni Resta, Jul 20 2013

Let r be a root of X^2 + 3*X + 1 in GF(prime(n)^2). Then n is in the sequence if and only if r^n = 1. - Robert Israel, Dec 24 2014

LINKS

Table of n, a(n) for n=1..11.

MAPLE

f:= proc(n)

  local p, m, r, t, F;

  p:= ithprime(n);

  if member(p mod 5, {1, 4}) then

  m:= igcd(n, p-1);

    r:= (numtheory:-msqrt(5, p)-3)/2 mod p;

    r &^ m mod p = 1

  else

    F:= GF(p, 2, t^2+3*t+1);

    m:= igcd(n, p^2-1);

    r:= F:-ConvertIn(t);

    F:-ConvertOut(F:-`^`(r, m)) = 1

  fi

end proc:

select(f, [$4 .. 10^5]); # Robert Israel, Dec 24 2014

MATHEMATICA

(* Mathematica's Fibonacci function is not used so as to speed up the search. *) fibo = {1, 1}; divFiboNPrimes = {}; Do[len = Length[fibo]; n = fibo[[len]] + fibo[[len - 1]]; fibo = Append[fibo, n]; If[Mod[n, Prime[i]] == 0, divFiboNPrimes = Append[divFiboNPrimes, i]], {i, 3, 10^7}]; divFiboNPrimes

PROG

(PARI) v=0; w=1; for(n=2, m, f=v+w; if(f%prime(n)==0, print1(n, ", ")); v=w; w=f)

CROSSREFS

Cf. A000040, A000045, A072123.

Sequence in context: A192771 A152963 A179698 * A205629 A253715 A205331

Adjacent sequences:  A075699 A075700 A075701 * A075703 A075704 A075705

KEYWORD

nonn

AUTHOR

Joseph L. Pe, Oct 02 2002

EXTENSIONS

Three more terms from Klaus Brockhaus, Oct 04 2002

a(7,8) = 2182656, 2582372 from Zak Seidov, Nov 03 2009

a(9)-a(11) from Giovanni Resta, Jul 20 2013

STATUS

approved

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Last modified August 23 11:46 EDT 2017. Contains 290995 sequences.