A282863 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 72*x^3 + (y-z)^3 is an even square.
1, 1, 1, 1, 2, 2, 3, 1, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 2, 2, 4, 3, 5, 2, 1, 3, 2, 4, 2, 3, 4, 2, 2, 3, 1, 3, 5, 3, 4, 1, 3, 4, 2, 2, 5, 2, 1, 3, 2, 3, 1, 4, 3, 2, 6, 3, 1, 4, 3, 3, 2, 4, 6, 1, 2, 6, 3, 3, 6, 3, 6, 2, 5, 3, 1, 6, 7, 5, 2, 4, 5
Offset: 0
Keywords
Examples
a(56) = 1 since 56 = 0^2 + 6^2 + 2^2 + 4^2 with 72*0^3 + (6-2)^3 = 8^2. a(120) = 1 since 120 = 4^2 + 2^2 + 10^2 + 0^2 with 72*4^3 + (2-10)^3 = 64^2. a(159) = 1 since 159 = 2^2 + 3^2 + 11^2 + 5^2 with 72*2^3 + (3-11)^3 = 8^2. a(1646) = 1 since 1646 = 5^2 + 10^2 + 0^2 + 39^2 with 72*5^3 + (10-0)^3 = 100^2. a(1784) = 1 since 1784 = 12^2 + 22^2 + 30^2 + 16^2 with 72*12^3 + (22-30)^3 = 352^2. a(3914) = 1 since 3914 = 2^2 + 45^2 + 21^2 + 38^2 with 72*2^3 + (45-21)^3 = 120^2. a(5864) = 1 since 5864 = 50^2 + 0^2 + 0^2 + 58^2 with 72*50^3 + (0-0)^3 = 3000^2.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
- Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
- Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
Programs
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Mathematica
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[72x^3+(y-z)^3]&&Mod[y-z,2]==0,r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r],{n,0,80}]
Comments