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

1,2

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)}|.

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(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.

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]

Crossrefs

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

Sequence in context: A080647 A324516 A181058 * A322788 A337082 A333505

Adjacent sequences: A301449 A301450 A301451 * A301453 A301454 A301455

Keyword

nonn

Author

Zhi-Wei Sun, Mar 21 2018

Status

approved