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%I #6 May 22 2018 10:04:52
%S 1,2,2,1,1,2,3,2,1,1,3,2,3,3,3,2,3,2,3,2,2,3,4,1,2,2,3,1,4,3,4,2,3,4,
%T 5,2,2,4,4,2,4,4,5,2,3,2,5,2,3,2,3,2,3,3,2,2,4,5,5,2,4,4,4,1,5,4,5,3,
%U 4,5,5,3,3,5,3,2,4,4,5,2,3,2,5,3,5,5,3,3,5,4,3,3,4,5,5,2,5,4,3,1
%N Number of nonnegative integers k such that n - k*L(k) is positive and squarefree, where L(k) denotes the k-th Lucas number A000032(k).
%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 9, 10, 24, 28, 64, 100, 104, 136, 153, 172, 176, 344, 496, 856, 928, 1036, 1084, 1216, 1860.
%H Zhi-Wei Sun, <a href="/A304945/b304945.txt">Table of n, a(n) for n = 1..100000</a>
%H 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 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 a(1) = 1 since 1 = 0*L(0) + 1 with 1 squarefree.
%e a(10) = 1 since 10 = 0*L(0) + 2*5 with 2*5 squarefree.
%e a(136) = 1 since 136 = 2*L(2) + 2*5*13 with 2*5*13 squarefree.
%e a(344) = 1 since 344 = 7*L(7) + 3*47 with 3*47 squarefree.
%e a(1036) = 1 since 1036 = 2*L(2) + 2*5*103 with 2*5*103 squarefree.
%e a(1860) = 1 since 1860 = 7*L(7) + 1657 with 1657 squarefree.
%t f[n_]:=f[n]=n*LucasL[n];
%t QQ[n_]:=QQ[n]=SquareFreeQ[n];
%t 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 Cf. A000032, A005117, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720, A304721, A304943.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, May 22 2018
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