A338162 Number of ways to write 4*n + 1 as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some k = 0,1,2,..., where x, y, z, w are nonnegative integers with z <= w.

1, 2, 3, 2, 4, 4, 2, 3, 6, 3, 7, 6, 5, 6, 7, 3, 8, 9, 5, 6, 8, 4, 8, 7, 4, 8, 11, 3, 7, 8, 6, 8, 13, 6, 6, 8, 6, 9, 11, 8, 10, 12, 7, 7, 12, 5, 14, 14, 7, 7, 13, 5, 13, 13, 5, 8, 13, 8, 10, 10, 7, 13, 10, 6, 9, 14, 9, 10, 15, 7, 10

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n >= 0. Moreover, if m > 1 has the form 2^a*(2*b+1), and either a is positive and even, or b is even, then m can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some positive integer k, where x, y, z, w are nonnegative integers.

We have verified the latter assertion in the conjecture for m up to 4*10^8.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..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(0) = 1, and 4*0 + 1 = 1^2 + 0^2 + 0^2 +0^2 with 1^2 + 7*0^2 = 2^0.
a(25) = 2, and 25 = 2^2 + 2^2 + 1^2 + 4^2 = 4^2 + 0^2 + 0^2 + 3^2
with 2^2 + 7*2^2 = 2^5 and 4^2 + 7*0^2 = 2^4.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[SQ[4n+1-x^2-y^2-z^2]&&PQ[x^2+7y^2], r=r+1], {x, 1, Sqrt[4n+1]}, {y, 0, Sqrt[4n+1-x^2]}, {z, 0, Sqrt[(4n+1-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 0, 70}]; tab

Crossrefs

Cf. A000079, A000118, A000290, A020670, A337082, A338094, A338095, A338096, A338139.

Sequence in context: A035583 A145178 A105079 * A215182 A214906 A274228

Adjacent sequences: A338159 A338160 A338161 * A338163 A338164 A338165

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2020

Status

approved