cp's OEIS Frontend

A304331 Number of integers k > 1 such that n - F(k) is a positive squarefree number, where F(k) denotes the k-th Fibonacci number A000045(k).

%I A304331 #17 May 14 2018 03:16:13
%S A304331 0,1,2,3,2,3,3,4,3,3,3,3,3,4,5,5,2,5,4,4,2,5,5,6,3,4,5,4,2,3,5,5,2,6,
%T A304331 6,7,4,5,6,6,4,6,6,7,4,4,6,5,4,4,5,4,2,5,5,7,3,5,5,8,4,5,6,6,4,5,6,7,
%U A304331 5,6,5,8,4,7,6,6,4,6,6,6,5,5,4,5,5,6,7,6,4,8
%N A304331 Number of integers k > 1 such that n - F(k) is a positive squarefree number, where F(k) denotes the k-th Fibonacci number A000045(k).
%C A304331 Conjecture: a(n) > 0 for all n > 1. In other words, every n = 2,3,... can be written as the sum of a positive Fibonacci number and a positive squarefree number.
%C A304331 This has been verified for n up to 10^10.
%C A304331 See also A304333 for a similar conjecture involving Lucas numbers.
%H A304331 Zhi-Wei Sun, <a href="/A304331/b304331.txt">Table of n, a(n) for n = 1..10000</a>
%H A304331 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
%H A304331 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 A304331 a(2) = 1 with 2 - F(2) = 1 squarefree.
%e A304331 a(53) = 2 with 53 - F(3) = 3*17 and 53 - F(9) = 19 both squarefree.
%t A304331 f[n_]:=f[n]=Fibonacci[n];
%t A304331 tab={};Do[r=0;k=2;Label[bb];If[f[k]>=n,Goto[aa]];If[SquareFreeQ[n-f[k]],r=r+1];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A304331 Cf. A000045, A005117, A304034, A304081, A304333.
%K A304331 nonn
%O A304331 1,3
%A A304331 _Zhi-Wei Sun_, May 11 2018