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%I #13 Apr 15 2021 04:45:45
%S 1,1,2,2,3,2,3,2,2,4,2,3,4,1,5,3,4,6,3,5,3,4,5,3,3,6,4,4,6,3,7,1,4,6,
%T 1,5,4,6,6,4,4,6,4,4,6,3,8,4,4,8,5,9,7,4,8,2,4,9,5,6,4,4,8,4,7,6,9,8,
%U 4,5,7,3,6,8,3,7,1,10,6,5,7,7,7,4,8,4,10,3,5,4,6,7,7,8,5,3,6,6,5,8
%N Number of ways to write n as x^2 + [y^2/4] + [z^4/6] with x,y,z positive integers, where [.] is the floor function.
%C Conjecture: a(n) > 0 for all n > 0.
%C We have verified a(n) > 0 for all n = 1..10^6.
%C See also A343387 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A343391/b343391.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(1) = 1 with 1 = 1^2 + [1^2/4] + [1^4/6].
%e a(2) = 1 with 2 = 1^2 + [2^2/4] + [1^4/6].
%e a(14) = 1 with 14 = 1^2 + [1^2/4] + [3^4/6].
%e a(32) = 1 with 32 = 4^2 + [8^2/4] + [1^4/6].
%e a(35) = 1 with 35 = 4^2 + [5^2/4] + [3^4/6].
%e a(77) = 1 with 77 = 8^2 + [1^2/4] + [3^4/6].
%e a(840) = 1 with 840 = 28^2 + [15^2/4] + [1^4/6].
%t SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/4]-Floor[y^4/6]],r=r+1],{x,1,Sqrt[4n+3]},{y,1,(6(n-Floor[x^2/4])+5)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y Cf. A000290, A000583, A343326, A343368, A343384, A343387, A343397, A343411.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 13 2021
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