A350860 - OEIS

%I #15 Jan 26 2022 13:02:38

%S 1,6,8,5,4,7,8,3,2,5,10,10,3,4,6,2,6,14,12,10,8,19,18,4,4,11,23,15,2,

%T 7,8,3,8,13,19,18,15,21,13,4,4,17,24,10,3,6,13,7,5,10,21,23,14,15,13,

%U 3,5,17,15,12,4,13,21,4,4,13,36,25,14,20,14,3,6,13,19,18,5,14,11,3,7,32,45,19,17,22,21,8,4,17,31

%N Number of ways to write n as w^4 + (x^4 + y^2 + z^2)/81, where w,x,y,z are nonnegative integers with y <= z.

%C Conjecture 1: a(n) > 0 for any nonnegative integer n. Also, every n = 0,1,2,... can be written as 4*w^4 + (x^4 + y^2 + z^2)/81 with w,x,y,z integers.

%C This implies that each nonnegative rational number can be written as w^4 + x^4 + y^2 + z^2 (or 4*w^4 + x^4 + y^2 + z^2) with w,x,y,z rational numbers.

%C Conjecture 2: For any positive integer c, there is a positive integer m such that every n = 0,1,2,... can be written as w^4 + (c^2*x^4 + y^2 + z^2)/m^4 with w,x,y,z integers.

%C This implies that for any positive integer c each nonnegative rational number can be written as w^4 + c^2*x^4 + y^2 + z^2 with w,x,y,z rational numbers.

%C Conjecture 3: For any positive odd number d, there is a positive integer m such that every n = 0,1,2,... can be written as 4*w^4 + (d^2*x^4 + y^2 + z^2)/m^4 with w,x,y,z integers.

%C This implies that for any positive odd number d each nonnegative rational number can be written as 4*w^4 + d^2*x^4 + y^2 + z^2 with w,x,y,z rational numbers.

%H Zhi-Wei Sun, <a href="/A350860/b350860.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.

%e a(8) = 2 with 8 = 0^4 + (0^4 + 18^2 + 18^2)/81 = 0^4 + (4^4 + 14^2 + 14^2)/81.

%e a(15) = 2 with 15 = 1^4 + (3^4 + 18^2 + 27^2)/81 = 1^4 + (5^4 + 5^2 + 22^2)/81.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[81(n-w^4)-x^4-y^2],r=r+1],{w,0,n^(1/4)},{x,0,3(n-w^4)^(1/4)},{y,0,Sqrt[(81(n-w^4)-x^4)/2]}];tab=Append[tab,r],{n,0,90}];Print[tab]

%Y Cf. A000290, A000583, A349942, A349943, A350857.

%K nonn,look

%O 0,2

%A _Zhi-Wei Sun_, Jan 19 2022