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

Author

Zhi-Wei Sun, Jul 22 2016

Keywords

• nonn

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.

Examples

			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|.
		

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.

Crossrefs

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

Programs

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}]