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A304689 Number of nonnegative integers k such that n - F(k)*F(k+1) is positive and squarefree, where F(k) denotes the k-th Fibonacci number A000045(k).

%I A304689 #7 May 17 2018 06:29:03
%S A304689 1,2,3,2,2,2,4,3,2,1,3,3,3,2,3,4,4,2,3,3,4,3,4,2,3,2,2,3,3,3,3,4,2,3,
%T A304689 4,4,4,3,4,3,5,3,5,4,4,3,4,4,3,2,2,3,5,3,3,2,5,3,5,2,5,4,4,2,2,4,4,4,
%U A304689 3,4,5,4,5,4,4,3,4,3,5,4,3,3,4,4,4,4,4,4,4,1,4,3,5,3,5,2,5,3,3,2
%N A304689 Number of nonnegative integers k such that n - F(k)*F(k+1) is positive and squarefree, where F(k) denotes the k-th Fibonacci number A000045(k).
%C A304689 Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 10, 90, 66690, 67452, 22756020.
%H A304689 Zhi-Wei Sun, <a href="/A304689/b304689.txt">Table of n, a(n) for n = 1..100000</a>
%H A304689 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
%H A304689 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e A304689 a(1) = 1 with 1 - F(0)*F(1) = 1 squarefree.
%e A304689 a(10) = 1 with 10 - F(0)*F(1) = 2*5 squarefree.
%e A304689 a(90) = 1 with 90 - F(1)*F(2) = 89 squarefree.
%e A304689 a(66690) = 1 with 66690 - F(10)*F(11) = 66690 - 55*89 = 5*17*727 squarefree.
%e A304689 a(67452) = 1 with 67452 - F(1)*F(2) = 37*1823 squarefree.
%e A304689 a(22756020) = 1 with 22756020 - F(2)*F(3) = 2*11378009 squarefree.
%t A304689 f[n_]:=f[n]=Fibonacci[n]*Fibonacci[n+1];
%t A304689 QQ[n_]:=QQ[n]=SquareFreeQ[n];
%t A304689 tab={};Do[r=0;k=0;Label[bb];If[f[k]>=n,Goto[aa]];If[QQ[n-f[k]],r=r+1];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A304689 Cf. A000045, A005117, A304034, A304081, A304331, A304333, A304522, A304523.
%K A304689 nonn
%O A304689 1,2
%A A304689 _Zhi-Wei Sun_, May 17 2018