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

%I A092330 #48 Sep 03 2019 03:06:29
%S A092330 1,1,1,3,5,29,152,136,2016,10959,26840,1056437,2495955,16311831,
%T A092330 102287808,1627690024,10021808981,25377192720,1085424779823,
%U A092330 2681584376185,17876295136009,113220181313816,1933742696582736
%N A092330 Fibonacci quotients: Fibonacci(p - Legendre(p|5))/p where p runs through the primes.
%C A092330 If p is prime then p divides fibonacci(p - Legendre(p|5)).
%C A092330 The result is known as the Fibonacci Quotient. - _John Blythe Dobson_, Sep 20 2014
%C A092330 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 A092330 Not to be confused with (Fibonacci(p) - Legendre(p|5))/p, which is A222361. - _Jonathan Sondow_, Dec 08 2017
%H A092330 Amiram Eldar, <a href="/A092330/b092330.txt">Table of n, a(n) for n = 1..647</a>
%H A092330 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 A092330 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 A092330 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number#Prime_divisors_of_Fibonacci_numbers">Prime divisors of Fibonacci numbers</a>
%H A092330 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 A092330 f:= proc(n) local p; p:= ithprime(n); combinat:-fibonacci(p - numtheory:-legendre(p,5))/p end proc:
%p A092330 seq(f(n),n=1..30); # _Robert Israel_, Sep 21 2014
%t A092330 a[n_] := With[{p = Prime[n]}, Fibonacci[p - KroneckerSymbol[p, 5]]/p];
%t A092330 Array[a, 23] (* _Jean-François Alcover_, Nov 25 2017 *)
%o A092330 (PARI) forprime (i=1,150,print1(fibonacci(i-kronecker(i,5))/i,","))
%Y A092330 Cf. A000045, A080891, A222361.
%K A092330 easy,nonn
%O A092330 1,4
%A A092330 Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 17 2004
%E A092330 Offset corrected by _Jonathan Sondow_, Dec 11 2017