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A349956 Number of ways to write n as x^2 + 2*y^2 + (z^4 + 4*w^4)/5 with x,y,z,w nonnegative integers.

%I A349956 #12 Dec 10 2021 07:10:36
%S A349956 1,2,2,2,3,2,2,2,2,3,3,1,3,4,1,2,4,5,5,6,4,3,6,3,3,6,4,3,5,4,2,5,4,6,
%T A349956 6,6,4,6,6,2,6,4,3,3,4,4,3,4,4,7,5,6,6,5,7,6,4,4,6,2,4,7,3,4,6,7,6,11,
%U A349956 9,4,11,6,7,9,7,3,9,8,1,7,7,8,10,11,7,10,9,5,8,9,8,3,7,6,3,6,5,10,10,10,11
%N A349956 Number of ways to write n as x^2 + 2*y^2 + (z^4 + 4*w^4)/5 with x,y,z,w nonnegative integers.
%C A349956 a(n) > 0 for all n <= 10^5.
%C A349956 Conjecture: If m is 5 or 65 or 85, then each n = 0,1,2,... can be written as x^2 + 2*y^2 + (z^4 + 4*w^4)/m with x,y,z,w nonnegative integers.
%C A349956 It seems that there are infinitely many positive squarefree numbers m (including 3, 5, 15, 23, 31, 33, 37, 55, 59, 67, 69, 71, 89, 93, 97, 111, 113, 115) such that every n = 0,1,2,... can be written as x^4 + 2*y^4 + (z^2 + 11*w^2)/m with x,y,z,w nonnegative integers.
%H A349956 Zhi-Wei Sun, <a href="/A349956/b349956.txt">Table of n, a(n) for n = 0..10000</a>
%H A349956 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 A349956 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 A349956 a(11) = 1 with 11 = 3^2 + 2*1^2 + (0^4 + 4*0^4)/5.
%e A349956 a(14) = 1 with 14 = 1^2 + 2*0^2 + (1^4 + 4*2^4)/5.
%e A349956 a(78) = 1 with 78 = 7^2 + 2*0^2 + (3^4 + 4*2^4)/5.
%e A349956 a(155) = 1 with 155 = 11^2 + 2*3^2 + (2^4 + 4*2^4)/5.
%e A349956 a(174) = 1 with 174 = 7^2 + 2*0^2 + (5^4 + 4*0^4)/5.
%t A349956 QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
%t A349956 tab={};Do[r=0;Do[If[QQ[5(n-2x^2-y^2)-4z^4],r=r+1],{x,0,Sqrt[n/2]},{y,0,Sqrt[n-2x^2]},{z,0,(5(n-2x^2-y^2)/4)^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A349956 Cf. A000290, A000583, A349942, A349945.
%K A349956 nonn
%O A349956 0,2
%A A349956 _Zhi-Wei Sun_, Dec 06 2021