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A282426 Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.

%I A282426 #10 Feb 15 2017 00:35:29
%S A282426 1,2,2,2,3,3,3,3,3,4,3,2,3,4,2,1,5,6,5,4,5,4,5,3,4,7,5,5,7,5,3,5,6,5,
%T A282426 6,4,5,5,6,4,7,7,6,9,6,5,6,5,4,6,4,4,7,6,6,6,5,2,6,3,4,6,6,5,6,6,6,10,
%U A282426 7,5
%N A282426 Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.
%C A282426 Conjecture: (i) a(n) > 0 for all n > 0.
%C A282426 (ii) Each n = 1,2,3,... can be written as x^4 + y^2 + z^2 + 2^k with x,y,z nonnegative integers and k among 0,1,2,3,4, 5.
%C A282426 (iii) Any positive integer n can be written as 4*x^4 + y^3 + z^2 + 3^k with k,x,y,z nonnegative integers.
%C A282426 We have verified parts (i) and (ii)-(iii) for n up to 2*10^7 and 10^7 respectively.
%C A282426 See arXiv:1701.05868 for more such conjectures.
%H A282426 Zhi-Wei Sun, <a href="/A282426/b282426.txt">Table of n, a(n) for n = 1..10000</a>
%H A282426 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e A282426 a(1) = 1 since 1 = 0^4 + 4*0^2 + 0^2 + 3^0.
%e A282426 a(16) = 1 since 16 = 0^4 + 4*1^2 + 3^2 + 3^1.
%e A282426 a(475) = 1 since 475 = 3^4 + 4*6^2 + 13^2 + 3^4.
%e A282426 a(556) = 1 since 556 = 0^4 + 4*0^2 + 23^2 + 3^3.
%e A282426 a(8641) = 1 since 8641 = 9^4 + 4*21^2 + 17^2 + 3^3.
%e A282426 a(52696) = 1 since 52696 = 12^4 + 4*87^2 + 41^2 + 3^1.
%t A282426 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A282426 Do[r=0;Do[If[SQ[n-x^4-4y^2-3^k],r=r+1],{k,0,Min[4,Log[3,n]]},{x,0,(n-3^k)^(1/4)},{y,0,Sqrt[(n-3^k-x^4)/4]}];Print[n," ",r];Continue,{n,1,80}]
%Y A282426 Cf. A000079, A000244, A000290, A000583, A280356.
%K A282426 nonn
%O A282426 1,2
%A A282426 _Zhi-Wei Sun_, Feb 14 2017