OFFSET
1,4
COMMENTS
If p is prime then p divides fibonacci(p - Legendre(p|5)).
The result is known as the Fibonacci Quotient. - John Blythe Dobson, Sep 20 2014
Legendre(p|5) = 1 if prime p == 1 or 4 mod 5, -1 if p == 2 or 3 mod 5, 0 if p = 5. - Robert Israel, Sep 21 2014
Not to be confused with (Fibonacci(p) - Legendre(p|5))/p, which is A222361. - Jonathan Sondow, Dec 08 2017
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..647
Zhi-Hong Sun and Zhi-Wei Sun, Fibonacci numbers and Fermat's last theorem, Acta Arithmetica 60(4) (1992), 371-388.
H. C. Williams, Some formulas concerning the fundamental unit of a real quadratic field, Discrete Mathematics, 92 (1991), 431-440.
Wikipedia, Prime divisors of Fibonacci numbers
Jianqiang Zhao, Finite Multiple zeta Values and Finite Euler Sums, arXiv preprint arXiv:1507.04917 [math.NT], 2015.
MAPLE
f:= proc(n) local p; p:= ithprime(n); combinat:-fibonacci(p - numtheory:-legendre(p, 5))/p end proc:
seq(f(n), n=1..30); # Robert Israel, Sep 21 2014
MATHEMATICA
a[n_] := With[{p = Prime[n]}, Fibonacci[p - KroneckerSymbol[p, 5]]/p];
Array[a, 23] (* Jean-François Alcover, Nov 25 2017 *)
PROG
(PARI) forprime (i=1, 150, print1(fibonacci(i-kronecker(i, 5))/i, ", "))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 17 2004
EXTENSIONS
Offset corrected by Jonathan Sondow, Dec 11 2017
STATUS
approved