A275300 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^3 with x + y + z a square, where x,y,z are integers with x >= |y| <= |z|, and w is a nonnegative integer.

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

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 5, 12, 20, 23, 31, 47, 71, 103, 148, 164.

The author proved in arXiv:1604.06723 that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x + y + z a square, where x,y,z,w are integers.

See also A275297, A275298, A275299 and A272620 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, arXiv:1604.06723 [math.GM], 2016.

Example

a(0) = 1 since 0 = 0^2 + 0^2 + 0^2 + 0^3 with 0 + 0 + 0 = 0^2 and 0 = 0 = 0.
a(5) = 1 since 5 = 2^2 + 0^2 + (-1)^2 + 0^3 with 2 + 0 + (-1) = 1^2 and 2 > 0 < |-1|.
a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^3 with 3 + (-1) + (-1) = 1^2 and 3 > |-1| = |-1|.
a(20) = 1 since 20 = 3^2 + 1^2 + (-3)^2 + 1^3 with 3 + 1 + (-3) = 1^2 and 3 > 1 < |-3|.
a(23) = 1 since 23 = 3^2 + (-2)^2 + 3^2 + 1^3 with 3 + (-2) + 3 = 2^2 and 3 > |-2| < 3.
a(31) = 1 since 31 = 5^2 + 1^2 + (-2)^2 + 1^3 with 5 + 1 + (-2) = 2^2 and 5 > 1 < |-2|.
a(47) = 1 since 47 = 6^2 + 1^2 + (-3)^2 + 1^3 with 6 + 1 + (-3) = 2^2 and 6 > 1 < |-3|.
a(71) = 1 since 71 = 6^2 + 3^2 + (-5)^2 + 1^3 with 6 + 3 + (-5) = 2^2 and 6 > 3 < |-5|.
a(103) = 1 since 103 = 7^2 + 2^2 + 7^2 + 1^3 with 7 + 2 + 7 = 4^2 and 7 > 2 < 7.
a(148) = 1 since 148 = 9^2 + (-2)^2 + (-6)^2 + 3^3 with 9 + (-2) + (-6) = 1^2 and 9 > |-2| < |-6|.
a(164) = 1 since 164 = 9^2 + 1^2 + (-9)^2 + 1^3 with 9 + 1 + (-9) = 1^2 and 9 > 1 < |-9|.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-w^3-y^2-z^2]&&SQ[Sqrt[n-w^3-y^2-z^2]+(-1)^i*y+(-1)^j*z], r=r+1], {w, 0, n^(1/3)}, {y, 0, Sqrt[(n-w^3)/3]}, {i, 0, Min[1, y]}, {z, y, Sqrt[n-w^3-2y^2]}, {j, 0, Min[1, z]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

Crossrefs

Cf. A000290, A000578, A271518, A272620, A275297, A275298, A275299.

Sequence in context: A239232 A346624 A153012 * A283833 A280759 A016651

Adjacent sequences: A275297 A275298 A275299 * A275301 A275302 A275303

Keyword

nonn

Author

Zhi-Wei Sun, Jul 22 2016

Status

approved