A338019 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 3*x + 10*y + 36*z a positive square, where x, y, z, w are nonnegative integers.

1, 1, 1, 1, 2, 1, 1, 0, 2, 2, 1, 2, 3, 3, 1, 1, 4, 1, 1, 2, 1, 3, 1, 0, 3, 4, 2, 1, 4, 4, 2, 1, 1, 3, 2, 2, 1, 5, 4, 0, 4, 4, 1, 1, 4, 3, 3, 1, 4, 3, 3, 4, 1, 4, 1, 2, 3, 3, 1, 4, 3, 3, 2, 1, 4, 2, 2, 2, 1, 1, 2, 1, 2, 3, 5, 1, 5, 5, 3, 2, 6, 4, 1, 6, 3, 5, 3, 1, 3, 7, 2, 2, 2, 7, 3, 1, 4, 1, 2, 2

Offset

1,5

Comments

Conjecture: a(n) > 0 if n is not divisible by 8. Moreover, a(n) = 0 if and only if n has the form 2^(4k+3)*m (k >= 0 and m = 1, 3, 5, 61).

We have verified this for n up to 5*10^6. See also A335624 for a similar conjecture.

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. See also arXiv:1604.06723 [math.NT].

Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Example

a(21) = 1, and 21 = 2^2 + 1^2 + 0^2 + 4^2 with 3*2 + 10*1 + 36*0 = 4^2.
a(98) = 1, and 98 = 6^2 + 7^2 + 3^2 + 2^2 with 3*6 + 10*7 + 36*3 = 14^2.
a(203) = 1, and 203 = 5^2 + 3^2 + 5^2 + 12^2 with 3*5 + 10*3 + 36*5 = 15^2.
a(760) = 1, and 760 = 0^2 + 18^2 + 20^2 + 6^2 with 3*0 + 10*18 + 36*20 = 30^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
TQ[n_]:=TQ[n]=n>0&&SQ[n];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&TQ[3x+10y+36z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}];
tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000118, A000290, A271518, A335624.

Sequence in context: A322285 A152487 A356894 * A058394 A353366 A122860

Adjacent sequences: A338016 A338017 A338018 * A338020 A338021 A338022

Keyword

nonn

Author

Zhi-Wei Sun, Oct 08 2020

Status

approved