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A343400 Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

%I A343400 #18 Apr 14 2021 05:20:18
%S A343400 1,2,3,4,4,5,4,6,4,9,5,7,8,6,9,6,7,9,7,6,9,7,8,7,7,10,6,9,11,9,12,8,9,
%T A343400 14,5,13,11,8,11,11,7,13,9,12,11,9,9,11,8,12,11,11,11,6,16,4,11,12,11,
%U A343400 13,12,6,10,9,8,17,8,12,11,10,8,10,12,10,8,11,12,12,13,7
%N A343400 Number of ways to write n as 3^x + [y^2/3] + [z^2/4], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.
%C A343400 Conjecture: a(n) > 0 for all n > 0.
%C A343400 We have verified a(n) > 0 for all n = 1..2*10^6.
%C A343400 The first indices n for which a(n) = 0 are 4051736, 7479656, 8592680, 9712160, 14039792, 16726256, 24914510. - _Giovanni Resta_, Apr 14 2021
%H A343400 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 A343400 a(2) = 2 with 2 = 3^0 + [1^2/3] + [2^2/4] = 3^0 + [2^2/3] + [1^2/4].
%e A343400 a(2942) = 2 with 2942 = 3^1 + [93^2/3] + [15^2/4] = 3^7 + [44^2/3] + [21^2/4].
%e A343400 a(627662) = 5 with 627662 - 3^0 = [330^2/3] + [1538^2/4] = [1042^2/3] + [1031^2/4] = [1318^2/3] + [441^2/4] = [1328^2/3] + [399^2/4] = [1352^2/3] + [271^2/4].
%e A343400 a(1103096) = 3 with 1103096 = 3^1 + [260^2/3] + [2079^2/4] = 3^1 + [508^2/3] + [2017^2/4] = 3^9 + [328^2/3] + [2047^2/4].
%e A343400 a(1694294) = 3 with 1694294 = 3^8 + [860^2/3] + [2401^2/4] = 3^8 + [928^2/3] + [2367^2/4] = 3^13 + [112^2/3] + [619^2/4].
%t A343400 PowQ[n_]:=PowQ[n]=IntegerQ[Log[3,n]];
%t A343400 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 A343400 Cf. A000290, A000244, A343387, A343391, A343397.
%K A343400 nonn
%O A343400 1,2
%A A343400 _Zhi-Wei Sun_, Apr 13 2021