A338095 Number of ways to write 2*n + 1 as x^2 + y^2 + z^2 + w^2 with x + y + 2*z a positive power of two, where x, y, z, w are nonnegative integers with x <= y.

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

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n >= 0. Moreover, any integer m > 10840 not congruent to 0 or 2 modulo 8 can be written as x^2 + y^2 + z^2 + w^2 with x + y + 2*z = 4^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 5*10^6. By Theorem 1.4(i) of the author's 2019 IJNT paper, any positive integer m can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w integers such that x + y + 2*z = 4^k for some nonnegative integer k.

See also A338094 and A338096 for similar conjectures.

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(3) = 1, and 2*3 + 1 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 1 + 2*1 = 2^2.
a(15) = 1, and 2*15 + 1 = 1^2 + 5^2 + 1^2 + 2^2 with 1 + 5 + 2*1 = 2^3.
a(69) = 1, and 2*69 + 1 = 7^2 + 9^2 + 0^2 + 3^2 with 7 + 9 + 2*0 = 2^4.
a(315) = 1, and 2*315 + 1 = 3^2 + 9^2 + 10^2 + 21^2 with 3 + 9 + 2*10 = 2^5.

Mathematica

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

Crossrefs

Cf. A000079, A000118, A000290, A000302, A279612, A338094, A338096, A338119, A338121.

Sequence in context: A054708 A112229 A087419 * A050979 A053450 A352533

Adjacent sequences: A338092 A338093 A338094 * A338096 A338097 A338098

Keyword

nonn

Author

Zhi-Wei Sun, Oct 09 2020

Status

approved