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A343397 Number of ways to write n as 2^x + [y^2/3] + [z^2/4] with x,y,z positive integers, where [.] is the floor function.

%I A343397 #18 Apr 14 2021 22:43:50
%S A343397 0,1,2,3,4,4,5,5,8,5,9,5,8,8,6,9,9,10,8,11,10,10,9,9,14,8,8,10,12,11,
%T A343397 6,14,13,10,12,13,15,11,13,9,20,6,12,17,13,13,10,11,17,12,11,13,15,14,
%U A343397 9,13,13,14,11,18,11,15,7,12,22,13,14,17,17,11,15,13,24,16,9,17,15,15,14,18
%N A343397 Number of ways to write n as 2^x + [y^2/3] + [z^2/4] with x,y,z positive integers, where [.] is the floor function.
%C A343397 Conjecture: a(n) > 0 for all n > 1.
%C A343397 We have verified a(n) > 0 for all n = 2..2*10^6.
%C A343397 Conjecture verified up to 10^10. - _Giovanni Resta_, Apr 14 2021
%H A343397 Zhi-Wei Sun, <a href="/A343397/b343397.txt">Table of n, a(n) for n = 1..10000</a>
%H A343397 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c]</a>, arXiv:1504.01608 [math.NT], 2015.
%e A343397 a(2) = 1 with 2 = 2^1 + [1^2/3] + [1^2/4].
%e A343397 a(3) = 2 with 3 = 2^1 + [1^2/3] + [2^2/4] = 2^1 + [2^2/3] + [1^2/4].
%t A343397 PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2,n]];
%t A343397 tab={};Do[r=0;Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]],r=r+1],{x,1,Sqrt[3n-1]},{y,1,Sqrt[4(n-Floor[x^2/3]-1)+1]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A343397 Cf. A000079, A000290, A343326, A343368, A343384, A343387, A343391, A343411.
%K A343397 nonn
%O A343397 1,3
%A A343397 _Zhi-Wei Sun_, Apr 13 2021