|
%I #22 Dec 25 2014 03:58:01
%S 2160,3048,27094,251712,505768,936240,2182656,2582372,487568736,
%T 1261336587,1424530096
%N n-th prime divides the n-th Fibonacci number.
%C a(12) > 2*10^9. - _Giovanni Resta_, Jul 20 2013
%C 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
%p f:= proc(n)
%p local p, m, r, t, F;
%p p:= ithprime(n);
%p if member(p mod 5, {1,4}) then
%p m:= igcd(n,p-1);
%p r:= (numtheory:-msqrt(5,p)-3)/2 mod p;
%p r &^ m mod p = 1
%p else
%p F:= GF(p,2,t^2+3*t+1);
%p m:= igcd(n,p^2-1);
%p r:= F:-ConvertIn(t);
%p F:-ConvertOut(F:-`^`(r,m)) = 1
%p fi
%p end proc:
%p select(f, [$4 .. 10^5]); # _Robert Israel_, Dec 24 2014
%t (* 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
%o (PARI) v=0; w=1; for(n=2,m,f=v+w; if(f%prime(n)==0,print1(n,",")); v=w; w=f)
%Y Cf. A000040, A000045, A072123.
%K nonn
%O 1,1
%A _Joseph L. Pe_, Oct 02 2002
%E Three more terms from _Klaus Brockhaus_, Oct 04 2002
%E a(7,8) = 2182656, 2582372 from _Zak Seidov_, Nov 03 2009
%E a(9)-a(11) from _Giovanni Resta_, Jul 20 2013
|
