A092330 Fibonacci quotients: Fibonacci(p - Legendre(p|5))/p where p runs through the primes.

1, 1, 1, 3, 5, 29, 152, 136, 2016, 10959, 26840, 1056437, 2495955, 16311831, 102287808, 1627690024, 10021808981, 25377192720, 1085424779823, 2681584376185, 17876295136009, 113220181313816, 1933742696582736

Offset: 1

Author

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 17 2004

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000045, A080891, A222361.

Programs

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 *)

PARI

forprime (i=1,150,print1(fibonacci(i-kronecker(i,5))/i,","))

Extensions

Offset corrected by Jonathan Sondow, Dec 11 2017