A279522 Number of ways to write n as w^2 + x^2 + y^2 + z^2 with w + 2*x + 3*y + 5*z a square, where w,x,y,z are nonnegative integers.

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 2, 2, 2, 1, 3, 5, 2, 2, 5, 4, 1, 1, 2, 4, 1, 0, 7, 3, 1, 1, 1, 4, 6, 4, 5, 4, 4, 3, 3, 5, 2, 3, 5, 4, 2, 1, 4, 8, 5, 1, 5, 12, 1, 1, 2, 6, 3, 4, 3, 3, 9, 1, 6, 4, 2, 3, 8, 8, 2, 3, 7, 7, 7, 3, 8, 14, 3, 2

Offset: 0

Author

Zhi-Wei Sun, Dec 14 2016

Keywords

• nonn

Comments

Conjecture: (i) a(n) = 0 if and only if n = 16^k*28 for some k = 0,1,2,....

(ii) For any positive integers a,b,c,d, there are infinitely many positive integers which cannot be written as w^2 + x^2 + y^2 + z^2 with a*w + b*x + c*y + d*z a square, where w,x,y,z are nonnegative integers.

Examples

			a(27) = 1 since 27 = 1^2 + 5^2 + 0^2 + 1^2 with 1 + 2*5 + 3*0 + 5*1 = 4^2.
a(31) = 1 since 31 = 1^2 + 2^2 + 5^2 + 1^2 with 1 + 2*2 + 3*5 + 5*1 = 5^2.
a(33) = 1 since 33 = 0^2 + 4^2 + 4^2 + 1^2 with 0 + 2*4 + 3*4 + 5*1 = 5^2.
a(52) = 1 since 52 = 4^2 + 6^2 + 0^2 + 0^2 with 4 + 2*6 + 3*0 + 5*0 = 4^2.
a(55) = 1 since 55 = 1^2 + 5^2 + 5^2 + 2^2 with 1 + 2*5 + 3*5 + 5*2 = 6^2.
a(56) = 1 since 56 = 0^2 + 4^2 + 6^2 + 2^2 with 0 + 2*4 + 3*6 + 5*2 = 6^2.
a(88) = 1 since 88 = 4^2 + 8^2 + 2^2 + 2^2 with 4 + 2*8 + 3*2 + 5*2 = 6^2.
a(137) = 1 since 137 = 10^2 + 6^2 + 1^2 + 0^2 with 10 + 2*6 + 3*1 + 5*0 = 5^2.
a(164) = 1 since 164 = 12^2 + 2^2 + 0^2 + 4^2 with 12 + 2*2 + 3*0 + 5*4 = 6^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..5000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016.

Crossrefs

Cf. A000118, A000290, A270969, A271518, A273404.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[Sqrt[n-x^2-y^2-z^2]+2x+3y+5z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,80}]