A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers.

0, 2, 2, 2, 2, 5, 3, 2, 4, 4, 4, 5, 5, 5, 6, 2, 4, 6, 5, 4, 9, 5, 4, 5, 5, 7, 10, 5, 6, 7, 8, 2, 6, 6, 7, 6, 9, 7, 10, 4, 6, 12, 3, 5, 10, 5, 6, 5, 5, 8, 9, 7, 7, 12, 5, 5, 13, 9, 6, 7, 8, 10, 13, 2, 6, 8, 10, 6, 15, 9, 9, 6, 10, 9, 12, 7, 8, 13, 6, 4

Offset: 1

Author

Zhi-Wei Sun, Mar 21 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1.

We have verified that a(n) > 0 for all n = 2..5*10^7.

It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)}| - |{d>0: d|n and d == 5,7 (mod 8)}|.

Examples

			a(2) = 2 since 2^2 = 2*4^0 + 0^2 + 2*1^2 and 2^2 = 3*4^0 + 1^2 + 2*0^2.
a(3) = 2 since 3^2 = 2*4^1 + 1^2 + 2*0^2 and 3^2 = 3*4^0 + 2^2 + 2*1^2.
a(5) = 2 since 5^2 = 2*4^1 + 3^2 + 2*2^2 and 5^2 = 3*4^0 + 2^2 + 2*3^2.
		

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. A000290, A000302, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391.

Programs

Mathematica

f[n_]:=f[n]=n/2^(IntegerExponent[n,2]);
OD[n_]:=OD[n]=Divisors[f[n]];
QQ[n_]:=QQ[n]=(n==0)||(n>0&&Sum[JacobiSymbol[-2,Part[OD[n],i]],{i,1,Length[OD[n]]}]!=0);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[QQ[n^2-m*4^k],Do[If[SQ[n^2-m*4^k-2x^2],r=r+1],{x,0,Sqrt[(n^2-m*4^k)/2]}]],{m,2,3},{k,0,Log[4,n^2/m]}];tab=Append[tab,r],{n,1,80}];Print[tab]