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

1,4

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

Cf. A000045, A080891, A222361.

Sequence in context: A240110 A228502 A215663 * A176951 A082716 A352295

Adjacent sequences: A092327 A092328 A092329 * A092331 A092332 A092333

Keyword

easy,nonn

Author

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

Extensions

Offset corrected by Jonathan Sondow, Dec 11 2017

Status

approved