A338095 Number of ways to write 2*n + 1 as x^2 + y^2 + z^2 + w^2 with x + y + 2*z a positive power of two, where x, y, z, w are nonnegative integers with x <= y.
1, 2, 4, 1, 2, 3, 4, 2, 5, 3, 4, 2, 3, 3, 4, 1, 2, 3, 4, 2, 6, 3, 3, 3, 4, 5, 6, 4, 6, 6, 5, 3, 9, 5, 4, 2, 4, 5, 6, 2, 5, 4, 5, 3, 6, 4, 4, 5, 5, 3, 6, 5, 4, 3, 4, 2, 6, 5, 4, 2, 3, 3, 7, 5, 4, 6, 5, 4, 7, 1, 2, 3, 6, 4, 3, 3, 5, 5, 4, 2, 6, 2, 5, 3, 2, 8, 7, 5, 6, 6, 6, 4, 10, 8, 7, 4, 4, 9, 8, 6, 10
Offset: 0
Keywords
Examples
a(3) = 1, and 2*3 + 1 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 1 + 2*1 = 2^2. a(15) = 1, and 2*15 + 1 = 1^2 + 5^2 + 1^2 + 2^2 with 1 + 5 + 2*1 = 2^3. a(69) = 1, and 2*69 + 1 = 7^2 + 9^2 + 0^2 + 3^2 with 7 + 9 + 2*0 = 2^4. a(315) = 1, and 2*315 + 1 = 3^2 + 9^2 + 10^2 + 21^2 with 3 + 9 + 2*10 = 2^5.
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. See also arXiv:1604.06723 [math.NT].
- Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
- Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.
Programs
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Mathematica
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; PQ[n_]:=PQ[n]=n>1&&IntegerQ[Log[2,n]]; tab={};Do[r=0;Do[If[SQ[2n+1-x^2-y^2-z^2]&&PQ[x+y+2z],r=r+1],{x,0,Sqrt[(2n+1)/2]},{y,x,Sqrt[2n+1-x^2]},{z,Boole[x+y==0],Sqrt[2n+1-x^2-y^2]}]; tab=Append[tab,r],{n,0,100}];Print[tab]
Comments