A337082 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x^2 + 4*y^2 - 7*x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

2, 2, 2, 2, 5, 5, 1, 2, 5, 4, 3, 3, 4, 7, 3, 2, 8, 8, 4, 6, 10, 6, 3, 5, 5, 9, 4, 2, 8, 10, 2, 2, 9, 4, 5, 6, 5, 7, 3, 4, 10, 10, 1, 4, 9, 6, 2, 3, 6, 8, 6, 4, 11, 12, 4, 7, 10, 5, 3, 5, 5, 9, 5, 2, 14, 16, 3, 9, 18, 9, 3, 8, 9, 11, 7, 5, 12, 14, 3, 6, 16, 11, 5, 12, 12, 10, 4, 6, 15, 17, 6, 5, 12, 9, 4, 5, 7, 12, 7, 7

Offset

1,1

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 2*x^2 + 4*y^2 - 7*x*y = 4^k for some positive integer k.

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

See also A338139 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(7) = 1, and 7 = 1^2 + 2^2 + 1^2 + 1^2 with 2*1^2 + 4*2^2 - 7*1*2 = 2^2.
a(43) = 1, and 43 = 4^2 + 1^2 + 1^2 + 5^2 with 2*4^2 + 4*1^2 - 7*4*1 = 2^3.
a(283) = 1, and 283 = 4^2 + 7^2 + 7^2 + 13^2 with 2*4^2 + 4*7^2 - 7*4*7 = 2^5.
a(2731) = 1, and 2731 = 5^2 + 7^2 + 16^2 + 49^2 with 2*5^2 + 4*7^2 - 7*5*7 = 2^0.
a(25475) = 1, and 25475 = 68^2 + 95^2 + 45^2 + 99^2 with 2*68^2 + 4*95^2 - 7*68*95 = 2^7.

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[2x^2+4*y^2-7*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, A000302, A279612, A282463, A338094, A338095, A338096, A338103, A338119, A338121, A338139.

Sequence in context: A181058 A301452 A322788 * A333505 A177333 A118486

Adjacent sequences: A337079 A337080 A337081 * A337083 A337084 A337085

Keyword

nonn

Author

Zhi-Wei Sun, Oct 12 2020

Status

approved