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

1,2

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.

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.

Example

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.

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]

Crossrefs

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

Sequence in context: A018838 A116982 A216621 * A214781 A214850 A236186

Adjacent sequences: A300445 A300446 A300447 * A300449 A300450 A300451

Keyword

nonn

Author

Zhi-Wei Sun, Mar 06 2018

Status

approved