A301391 Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y even such that x^2 - (6*y)^2 = 4^k for some k = 0,1,2,....

1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 3, 4, 1, 1, 8, 2, 2, 2, 3, 2, 6, 1, 2, 2, 1, 1, 11, 3, 2, 4, 4, 3, 3, 1, 6, 10, 6, 2, 7, 2, 3, 2, 6, 3, 8, 2, 7, 7, 2, 1, 11, 4, 4, 2, 2, 1, 6, 1, 3, 11, 3, 3, 16, 3, 5, 4, 8, 5, 2, 3, 11, 5, 8, 1

Offset: 1

Author

Zhi-Wei Sun, Mar 20 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 11, 13, 19, 2^k*m (k = 0,1,2,... and m = 1, 5, 7, 31).

We have verified a(n) > 0 for all n = 1..10^7.

See also A301376 for a similar conjecture.

Examples

			a(2) = 1 since 2^2 = 2^2 + 0^2 + 0^2 + 0^2 with 2^2 - (6*0)^2 = 4^1.
a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4^2 - (6*0)^2 = 4^2.
a(7) = 1 since 7^2 = 2^2 + 0^2 + 3^2 + 6^2 with 2^2 - (6*0)^2 = 4^1.
a(11) = 1 since 11 = 2^2 + 0^2 + 6^2 + 9^2 with 2^2 - (6*0)^2 = 4^1.
a(13) = 1 since 13 = 4^2 + 0^2 + 3^2 + 12^2 with 4^2 - (6*0)^2 = 4^2.
a(19) = 1 since 19 = 1^2 + 0^2 + 6^2 + 18^2 with 1^2 - (6*0)^2 = 4^0.
a(31) = 1 since 31^2 = 20^2 + 2^2 + 14^2 + 19^2 with 20^2 - (6*2)^2 = 4^4.
a(75) = 2 since 75^2 = 68^2 + 10^2 + 1^2 + 30^2 = 68^2 + 10^2 + 15^2 + 26^2 with 68^2 - (6*10)^2 = 4^5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..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], 2017-2018.

Crossrefs

Cf. A000118, A000290, A000302, A299537, A299794, A299924, A300219, A300396, A300441, A300510, A301376.

Programs

Mathematica

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4^k+144m^2]&&QQ[n^2-4^k-148m^2], Do[If[SQ[n^2-(4^k+148m^2)-z^2],r=r+1],{z,0,Sqrt[(n^2-4^k-148m^2)/2]}]],{k,0,Log[2,n]},{m,0,Sqrt[(n^2-4^k)/148]}];tab=Append[tab,r],{n,1,80}];Print[tab]