A282972 Number of ways to write n as 4*x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 79*y^2 - 220*y*z + 205*z^2 is a square.
1, 1, 1, 1, 2, 2, 3, 1, 2, 4, 3, 2, 2, 4, 4, 2, 2, 3, 5, 2, 2, 4, 4, 2, 3, 3, 3, 2, 2, 3, 2, 2, 1, 4, 2, 1, 4, 2, 3, 1, 4, 3, 2, 1, 3, 5, 2, 1, 3, 6, 3, 2, 2, 5, 5, 2, 4, 3, 4, 2, 3, 5, 2, 2, 2, 6, 5, 2, 4, 5, 6, 1, 5, 6, 5, 4, 5, 5, 6, 2, 4
Offset: 0
Keywords
Examples
a(2) = 1 since 2 = 4*0^4 + 1^2 + 1^2 + 0^2 with 79*1^2 - 220*1*1 + 205*1^2 = 8^2. a(35) = 1 since 35 = 4*0^4 + 3^2 + 1^2 + 5^2 with 79*3^2 - 220*3*1 + 205*1^2 = 16^2. a(119) = 1 since 119 = 4*1^4 + 9^2 + 3^2 + 5^2 with 79*9^2 - 220*9*3 + 205*3^2 = 48^2. a(124) = 1 since 124 = 4*1^4 + 4^2 + 2^2 + 10^2 with 79*4^2 - 220*4*2 + 205*2^2 = 18^2. a(1564) = 1 since 1564 = 4*3^4 + 14^2 + 30^2 + 12^2 with 79*14^2 - 220*14*30 + 205*30^2 = 328^2. a(4619) = 1 since 4619 = 4*2^4 + 51^2 + 27^2 + 35^2 with 79*51^2 - 220*51*27 + 205*27^2 = 228^2. a(6127) = 1 since 6127 = 4*5^4 + 49^2 + 35^2 + 1^2 with 79*49^2 - 220*49*35 + 205*35^2 = 252^2. a(7119) = 1 since 7119 = 4*1^4 + 51^2 + 17^2 + 65^2 with 79*51^2 - 220*51*17 + 205*17^2 = 272^2. a(9087) = 1 since 9087 = 4*3^4 + 61^2 + 71^2 + 1^2 with 79*61^2 - 220*61*71 + 205*71^2 = 612^2.
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-4x^4-y^2-z^2]&&SQ[79y^2-220*y*z+205z^2],r=r+1],{x,0,(n/4)^(1/4)},{y,0,Sqrt[n-4x^4]},{z,0,Sqrt[n-4x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]
Comments