A300448 Number of the integers 4^k*m with k >= 0 and m = 1, 2, 5 such that n^2 - 4^k*m can be written as the sum of two squares.

1, 2, 4, 2, 4, 5, 4, 2, 7, 5, 6, 5, 6, 5, 5, 2, 5, 7, 6, 5, 10, 7, 5, 5, 7, 7, 6, 5, 6, 6, 7, 2, 10, 6, 8, 7, 7, 6, 6, 5, 7, 11, 5, 7, 10, 5, 6, 5, 8, 8, 9, 7, 8, 6, 6, 5, 10, 7, 5, 6, 5, 8, 7, 2, 5, 10, 8, 6, 10, 8, 7, 7, 11, 7, 7, 6, 10, 7, 5, 5

Offset: 1

Author

Zhi-Wei Sun, Mar 06 2018

Keywords

• nonn

Comments

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

Conjecture 2: Each positive square n^2 can be written as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5 17.

Conjecture 3: For each n = 1,2,3,... we can write n^2 as 4^k*m + x^2 + y^2 with k,x,y nonnegative integers and m among 1, 5, 65.

Conjecture 3 implies that A300441(n) > 0 for all n > 0 since 4*A001353(0)^2 + 1 = 1, 4*A001353(1)^2 + 1 = 5 and 4*A001353(2)^2 + 1 = 65.

We have verified Conjectures 1-3 for all n = 1..2*10^7.

Examples

			a(1) = 1 since 1^2 - 4^0*1 = 0 = 0^2 + 0^2.
a(2) = 2 since 2^2 - 4^0*2 = 2 = 1^2 + 1^2 and 2^2 - 4*1 = 0 = 0^2 + 0^2.
a(4) = 2 since 4^2 - 4^1*2 = 8 = 2^2 + 2^2 and 4^2 - 4^2*1 = 0 = 0^2 + 0^2.
		

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. A000118, A000302, A001481, A271518, A281976, A299924, A299537, A299794, A300362, A300441, A300396.

Programs

Mathematica

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1,1,0],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
u[1]=1; u[2]=2; u[3]=5;
tab={};Do[r=0;Do[If[QQ[n^2-4^k*u[m]],r=r+1],{m,1,3},{k,0,Log[4,n^2/u[m]]}];tab=Append[tab,r],{n,1,80}];Print[tab]