A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26*y^2 - 11*x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

1, 2, 2, 2, 3, 4, 2, 2, 4, 5, 3, 4, 3, 4, 3, 2, 4, 6, 3, 5, 6, 4, 2, 4, 4, 5, 4, 4, 4, 6, 2, 2, 7, 5, 3, 6, 5, 4, 3, 5, 7, 8, 1, 4, 8, 4, 2, 4, 5, 6, 4, 5, 5, 6, 4, 4, 8, 5, 2, 6, 4, 3, 3, 2, 8, 11, 3, 5, 11, 6, 1, 6, 8, 7, 5, 4, 6, 5, 1, 5, 10, 10, 5, 9, 8, 5, 4, 4, 8, 14, 5, 5, 8, 4, 4, 4, 6, 7, 5, 7

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0. Moreover, any positive integer n congruent to 1 or 2 modulo 4 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^2 + 26*y^2 - 11*x*y = 4^k for some nonnegative integer k.

We have verified this for all n = 1..10^8.

See also A337082 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, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893.  See also arXiv:1701.05868 [math.NT].

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

Example

a(1) = 1, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1^2 + 26*0^2 - 11*1*0 = 2^0.
a(43) = 1, and 43 = 1^2 + 1^2 + 4^2 + 5^2 with 1^2 + 26*1^2 - 11*1*1 = 2^4.
a(6547) = 1, and 6547 = 17^2 + 1^2 + 4^2 + 79^2 with 17^2 + 26*1^2 - 11*17*1 = 2^7.
a(11843) = 1, and 11843 = 3^2 + 1^2 + 13^2 + 108^2 with 3^2 + 26*1^2 - 11*3*1 = 2^1.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=n>0&&IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&PQ[x^2+26*y^2-11*x*y], r=r+1], {x, 0, Sqrt[n]}, {y, Boole[x==0], Sqrt[n-x^2]}, {z, 0, Sqrt[(n-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 1, 100}]; tab

Crossrefs

Cf. A000079, A000118, A000290, A279612, A282463, A338094, A338095, A338096, A338103, A338119, A338121.

Sequence in context: A282933 A328576 A052275 * A244798 A308253 A286614

Adjacent sequences: A338136 A338137 A338138 * A338140 A338141 A338142

Keyword

nonn

Author

Zhi-Wei Sun, Oct 12 2020

Status

approved