cp's OEIS Frontend

A304523 Number of ordered ways to write n as the sum of a Lucas number (A000032) and a positive odd squarefree number.

%I A304523 #21 Nov 04 2018 20:33:35
%S A304523 0,1,1,2,2,2,2,3,2,2,1,3,1,4,2,3,2,4,3,3,3,4,3,4,3,3,1,2,1,4,2,4,3,4,
%T A304523 3,4,3,3,3,5,3,5,2,5,2,4,2,5,2,5,2,4,2,5,3,2,3,6,3,5,3,6,2,5,2,5,1,6,
%U A304523 3,5,3,5,3,3,3,5,3,4,3,6,3,4,3,5,2,5,4,5,4,6
%N A304523 Number of ordered ways to write n as the sum of a Lucas number (A000032) and a positive odd squarefree number.
%C A304523 Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 11, 13, 27, 29, 67, 139, 193, 247, 851.
%C A304523 It has been verified that a(n) > 0 for all n = 2..5*10^9.
%C A304523 See also A304331, A304333 and A304522 for similar conjectures.
%H A304523 Zhi-Wei Sun, <a href="/A304523/b304523.txt">Table of n, a(n) for n = 1..100000</a>
%H A304523 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 A304523 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 A304523 a(3) = 1 since 3 = A000032(0) + 1 with 1 odd and squarefree.
%e A304523 a(27) = 1 since 27 = A000032(3) + 23 with 23 odd and squarefree.
%e A304523 a(29) = 1 since 29 = A000032(6) + 11 with 11 odd and squarefree.
%e A304523 a(67) = 1 since 67 = A000032(0) + 5*13 with 5*13 odd and squarefree.
%e A304523 a(247) = 1 since 247 = A000032(6) + 229 with 229 odd and squarefree.
%e A304523 a(851) = 1 since 851 = A000032(0) + 3*283 with 3*283 odd and squarefree.
%t A304523 f[n_]:=f[n]=LucasL[n];
%t A304523 QQ[n_]:=QQ[n]=n>0&&Mod[n,2]==1&&SquareFreeQ[n];
%t A304523 tab={};Do[r=0;k=0;Label[bb];If[k>0&&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,1,90}];Print[tab]
%Y A304523 Cf. A000032, A005117, A304034, A304081, A304331, A304333, A304522.
%K A304523 nonn
%O A304523 1,4
%A A304523 _Zhi-Wei Sun_, May 13 2018