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A349661 Number of ways to write n as x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.

%I A349661 #66 Dec 09 2021 08:15:40
%S A349661 1,3,3,2,4,5,2,2,4,5,6,4,3,6,4,1,6,7,4,6,8,5,1,4,6,10,10,3,6,10,2,3,8,
%T A349661 6,10,10,5,6,4,5,12,14,6,5,9,6,2,3,6,12,14,7,5,8,2,7,14,6,9,9,5,9,4,2,
%U A349661 10,15,7,7,8,7,3,5,5,7,14,5,9,9,1,4,11,8,11,13,7,13,7,2,11,17,12,8,5,6,7,5,7,11,12,8
%N A349661 Number of ways to write n as x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.
%C A349661 Conjecture: a(n) > 0 for all n > 0.
%C A349661 This has been verified for n up to 10^6.
%C A349661 As (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2, the conjecture gives a new refinement of Lagrange's four-square theorem.
%C A349661 See also A350012 for a similar conjecture.
%H A349661 Zhi-Wei Sun, <a href="/A349661/b349661.txt">Table of n, a(n) for n = 1..10000</a>
%H A349661 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167--190.
%H A349661 Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893.  See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.
%H A349661 Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.
%e A349661 a(1) = 1 with 1 = 0^4 + 0^2 + (1^2 + 4^0)/2.
%e A349661 a(23) = 1 with 23 = 1^4 + 3^2 + (5^2 + 4^0)/2.
%e A349661 a(79) = 1 with 79 = 1^4 + 2^2 + (12^2 + 4^1)/2.
%e A349661 a(1199) = 1 with 1199 = 5^4 + 18^2 + (22^2 + 4^2)/2.
%e A349661 a(3679) = 1 with 3679 = 5^4 + 2^2 + (78^2 + 4^2)/2.
%e A349661 a(6079) = 1 with 6079 = 3^4 + 42^2 + (92^2 + 4^1)/2.
%e A349661 a(33439) = 1 with 33439 = 1^4 + 175^2 + (75^2 + 4^0)/2.
%e A349661 a(50399) = 1 with 50399 = 13^4 + 135^2 + (85^2 + 4^0)/2.
%e A349661 a(207439) = 1 with 207439 = 1^4 + 142^2 + (612^2 + 4^1)/2.
%t A349661 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A349661 tab={};Do[r=0;Do[If[SQ[2(n-x^4-y^2)-4^z],r=r+1],{x,0,(n-1)^(1/4)},{y,0,Sqrt[n-1-x^4]},{z,0,Log[4,2(n-x^4-y^2)]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A349661 Cf. A000118, A000290, A000302, A000583, A349957, A349992, A350012.
%K A349661 nonn
%O A349661 1,2
%A A349661 _Zhi-Wei Sun_, Dec 08 2021