cp's OEIS Frontend

A304943 Number of ways to write n as the sum of a positive tribonacci number (A000073) and a positive odd squarefree number.

%I A304943 #7 May 22 2018 09:54:50
%S A304943 0,1,1,1,2,1,2,2,2,1,1,2,1,3,2,2,2,3,2,3,2,2,2,3,3,2,2,2,1,3,2,2,2,2,
%T A304943 3,3,3,2,3,2,3,3,3,3,4,2,3,3,2,2,2,2,2,3,4,2,4,2,4,3,4,2,4,2,3,3,3,3,
%U A304943 2,2,3,3,3,3,4,1,3,3,3,3,4,2,3,4,3,4,3,2,3,3,4,4,3,3,4,4,4,4,3,3
%N A304943 Number of ways to write n as the sum of a positive tribonacci number (A000073) and a positive odd squarefree number.
%C A304943 Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 6, 10, 11, 13, 29, 76, 1332, 25249.
%H A304943 Zhi-Wei Sun, <a href="/A304943/b304943.txt">Table of n, a(n) for n = 1..100000</a>
%H A304943 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 A304943 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 A304943 a(2) = 1 with 2 = 1 + 1, where 1 = A000073(2) = A000073(3) is a positive tribonacci number, and 1 is also odd and squarefree.
%e A304943 a(29) = 1 since 29 = A000073(8) + 5 with 5 odd and squarefree.
%e A304943 a(76) = 1 since 76 = A000073(6) + 3*23 with 3*23 odd and squarefree.
%e A304943 a(1332) = 1 since 1332 = A000073(7) + 1319 with 1319 odd and squarefree.
%e A304943 a(25249) = 1 since 25249 = A000073(4) + 25247 with 25247 odd and squarefree.
%t A304943 f[0]=0;f[1]=0;f[2]=1;
%t A304943 f[n_]:=f[n]=f[n-1]+f[n-2]+f[n-3];
%t A304943 QQ[n_]:=QQ[n]=Mod[n,2]==1&&SquareFreeQ[n];
%t A304943 tab={};Do[r=0;k=3;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 A304943 Cf. A000073, A005117, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720, A304721.
%K A304943 nonn
%O A304943 1,5
%A A304943 _Zhi-Wei Sun_, May 22 2018