A350021 Number of ways to write n as w^4 + x^2 + y^2 + z^2 with x - y a power of two (including 2^0 = 1).

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

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0.

This is a new refinement of Lagrange's four-square theorem, and we have verified it for n up to 10^6.

If x - y = 2^k, then x^2 + y^2 = ((x+y)^2 + (2^k)^2)/2 and x + y >= 2^k. So the above conjecture implies the conjecture in A349661.

In his 2017 JNT paper, the author proved that each n = 0,1,2,... can be written as w^4 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers.

In his 2019 IJNT paper, the author proved that any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that x - y is a power of two (including 2^0 = 1).

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.

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, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Example

a(3) = 1 with 3 = 1^4 + 1^2 + 0^2 + 1^2 and 1 - 0 = 2^0.
a(4) = 1 with 4 = 0^4 + 2^2 + 0^2 + 0^2 and  2 - 0 = 2^1.
a(7) = 1 with 7 = 1^4 + 2^2 + 1^2 + 1^2 and 2 - 1 = 2^0.
a(8) = 1 with 8 = 0^4 + 2^2 + 0^2 + 2^2 and 2 - 0 = 2^1.
a(12) = 1 with 12 = 1^4 + 3^2 + 1^2 + 1^2 and 3 - 1 = 2^1.
a(19) = 1 with 19 = 0^4 + 3^2 + 1^2 + 3^2 and 3 - 1 = 2^1.
a(28) = 1 with 28 = 1^4 + 5^2 + 1^2 + 1^2 and 5 - 1 = 2^2.
a(47) = 1 with 47 = 1^4 + 3^2 + 1^2 + 6^2 and 3 - 1 = 2^1.
a(55) = 1 with 55 = 1^4 + 2^2 + 1^2 + 7^2 and 2 - 1 = 2^0.
a(88) = 1 with 88 = 0^4 + 6^2 + 4^2 + 6^2 and 6 - 4 = 2^1.
a(103) = 1 with 103 = 3^4 + 3^2 + 2^2 + 3^2 and 3 - 2 = 2^0.
a(193) = 1 with 193 = 2^4 + 8^2 + 7^2 + 8^2 and 8 - 7 = 2^0.
a(439) = 1 with 439 = 3^4 + 5^2 + 3^2 + 18^2 and 5 - 3 = 2^1.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PowQ[n_]:=PowQ[n]=IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[SQ[n-w^4-x^2-y^2]&&PowQ[y-x], r=r+1], {w, 0, (n-1)^(1/4)}, {x, 0, Sqrt[(n-w^4)/2]}, {y, x+1, Sqrt[n-w^4-x^2]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000118, A000290, A000583, A279612, A300708, A338094, A349661, A350012.

Sequence in context: A083293 A350912 A055370 * A176388 A282494 A156609

Adjacent sequences: A350018 A350019 A350020 * A350022 A350023 A350024

Keyword

nonn

Author

Zhi-Wei Sun, Dec 08 2021

Status

approved