A092330 - OEIS

%I #48 Sep 03 2019 03:06:29

%S 1,1,1,3,5,29,152,136,2016,10959,26840,1056437,2495955,16311831,

%T 102287808,1627690024,10021808981,25377192720,1085424779823,

%U 2681584376185,17876295136009,113220181313816,1933742696582736

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

%C If p is prime then p divides fibonacci(p - Legendre(p|5)).

%C The result is known as the Fibonacci Quotient. - _John Blythe Dobson_, Sep 20 2014

%C 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

%C Not to be confused with (Fibonacci(p) - Legendre(p|5))/p, which is A222361. - _Jonathan Sondow_, Dec 08 2017

%H Amiram Eldar, <a href="/A092330/b092330.txt">Table of n, a(n) for n = 1..647</a>

%H Zhi-Hong Sun and Zhi-Wei Sun, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf">Fibonacci numbers and Fermat's last theorem</a>, Acta Arithmetica 60(4) (1992), 371-388.

%H H. C. Williams, <a href="http://dx.doi.org/10.1016/0012-365X(91)90298-G">Some formulas concerning the fundamental unit of a real quadratic field,</a> Discrete Mathematics, 92 (1991), 431-440.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number#Prime_divisors_of_Fibonacci_numbers">Prime divisors of Fibonacci numbers</a>

%H Jianqiang Zhao, <a href="http://arxiv.org/abs/1507.04917">Finite Multiple zeta Values and Finite Euler Sums</a>, arXiv preprint arXiv:1507.04917 [math.NT], 2015.

%p f:= proc(n) local p; p:= ithprime(n); combinat:-fibonacci(p - numtheory:-legendre(p,5))/p end proc:

%p seq(f(n),n=1..30); # _Robert Israel_, Sep 21 2014

%t a[n_] := With[{p = Prime[n]}, Fibonacci[p - KroneckerSymbol[p, 5]]/p];

%t Array[a, 23] (* _Jean-François Alcover_, Nov 25 2017 *)

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

%Y Cf. A000045, A080891, A222361.

%K easy,nonn

%O 1,4

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

%E Offset corrected by _Jonathan Sondow_, Dec 11 2017