|
|
A075702
|
|
n-th prime divides the n-th Fibonacci number.
|
|
5
|
|
|
2160, 3048, 27094, 251712, 505768, 936240, 2182656, 2582372, 487568736, 1261336587, 1424530096
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
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
|
|
|
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:
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(7,8) = 2182656, 2582372 from Zak Seidov, Nov 03 2009
|
|
STATUS
|
approved
|
|
|
|