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A343384 Number of ways to write n as [a^3/3] + [b^3/4] + [c^3/5] + [d^6/6] with a,b,c,d positive integers, where [.] is the floor function.

%I A343384 #20 Apr 28 2021 04:17:13
%S A343384 1,1,2,2,1,2,1,3,1,3,2,3,4,3,4,3,5,4,3,4,3,5,3,4,4,3,6,5,5,2,4,5,3,6,
%T A343384 3,3,4,6,5,2,4,5,4,7,3,5,4,4,5,3,3,4,7,6,3,6,4,5,6,5,1,3,7,3,5,3,5,3,
%U A343384 8,4,3,2,6,3,6,4,6,4,6,5,5,1,5,5,7,4,7,6,4,6,5,2,2,5,5,5,5,6,3,7,7
%N A343384 Number of ways to write n as [a^3/3] + [b^3/4] + [c^3/5] + [d^6/6] with a,b,c,d positive integers, where [.] is the floor function.
%C A343384 3-4-5-6 Conjecture: a(n) > 0 for all n >= 0.
%C A343384 We have verified a(n) > 0 for all n = 0..10^6.
%C A343384 Conjecture verified up to 2*10^9. - _Giovanni Resta_, Apr 28 2021
%H A343384 Zhi-Wei Sun, <a href="/A343384/b343384.txt">Table of n, a(n) for n = 0..10000</a>
%H A343384 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c)</a>, arXiv:1504.01608 [math.NT], 2015.
%e A343384 a(0) = 1 with 0 = [1^3/3] + [1^3/4] + [1^3/5] + [1^6/6].
%e A343384 a(1) = 1 with 1 = [1^3/3] + [1^3/4] + [2^3/5] + [1^6/6].
%e A343384 a(4) = 1 with 4 = [2^3/3] + [2^3/4] + [1^3/5] + [1^6/6].
%e A343384 a(6) = 1 with 6 = [1^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
%e A343384 a(8) = 1 with 8 = [2^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
%e A343384 a(60) = 1 with 60 = [3^3/3] + [4^3/4] + [5^3/5] + [2^6/6].
%e A343384 a(81) = 1 with 81 = [2^3/3] + [6^3/4] + [5^3/5] + [1^6/6].
%e A343384 a(300) = 1 with 300 = [7^3/3] + [5^3/4] + [9^3/5] + [2^6/6].
%e A343384 a(4434) = 1 with 4434 = [11^3/3] + [4^3/4] + [19^3/5] + [5^6/6].
%t A343384 CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
%t A343384 tab={};Do[r=0;Do[If[CQ[3(n-Floor[x^6/6]-Floor[y^3/5]-Floor[z^3/4])+s],r=r+1],{s,0,2},{x,1,(6n+5)^(1/6)},{y,1,(5(n-Floor[x^6/6])+4)^(1/3)},{z,1,(4(n-Floor[x^6/6]-Floor[y^3/5])+3)^(1/3)}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A343384 Cf. A000578, A001014, A343326, A343368, A343387.
%K A343384 nonn
%O A343384 0,3
%A A343384 _Zhi-Wei Sun_, Apr 13 2021