A273429 Number of ordered ways to write n as x^6 + y^2 + z^2 + w^2, where x,y,z,w are nonnegative integers with y <= z <= w.

1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 2, 2, 2, 2, 1, 1, 3, 4, 3, 2, 2, 2, 1, 1, 3, 4, 4, 2, 2, 3, 1, 1, 3, 4, 3, 3, 3, 3, 2, 1, 4, 4, 2, 2, 3, 3, 1, 1, 3, 5, 5, 3, 3, 5, 3, 1, 3, 3, 3, 2, 2, 4, 2, 2, 5, 7, 5, 4, 5, 4, 1, 3, 6, 6, 6, 4, 4, 4, 1, 2

Offset: 0

Author

Zhi-Wei Sun, May 22 2016

Keywords

• nonn

Comments

The author proved in arXiv:1604.06723 that for each c = 1, 4 any natural number can be written as c*x^6 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers. Thus a(n) > 0 for all n = 0,1,2,....

We note that a(n) = 1 for the following values of n not divisible by 2^6: 7, 8, 15, 16, 23, 24, 31, 32, 40, 47, 48, 56, 71, 79, 92, 112, 143, 176, 191, 240, 304, 368, 560, 624, 688, 752, 1072, 1136, 1456, 1520, 1840, 1904, 2608, 2672, 3760, 3824, 6512, 6896.

For more conjectural refinements of Lagrange's four-square theorem, one may consult the author's preprint arXiv:1604.06723.

Examples

			a(7) = 1 since 7 = 1^6 + 1^2 + 1^2 + 2^2 with 1 = 1 < 2.
a(8) = 1 since 8 = 0^6 + 0^2 + 2^2 + 2^2 with 0 < 2 = 2.
a(15) = 1 since 15 = 1^6 + 1^2 + 2^2 + 3^2 with 1 < 2 < 3.
a(16) = 1 since 16 = 0^6 + 0^2 + 0^2 + 4^2 with 0 = 0 < 4.
a(56) = 1 since 56 = 0^6 + 2^2 + 4^2 + 6^2 with 2 < 4 < 6.
a(71) = 1 since 71 = 1^6 + 3^2 + 5^2 + 6^2 with 3 < 5 < 6.
a(79) = 1 since 79 = 1^6 + 2^2 + 5^2 + 7^2 with 2 < 5 < 7.
a(92) = 1 since 92 = 1^6 + 1^2 + 3^2 + 9^2 with 1 < 3 < 9.
a(143) = 1 since 143 = 1^6 + 5^2 + 6^2 + 9^2 with 5 < 6 < 9.
a(191) = 1 since 191 = 1^6 + 3^2 + 9^2 + 10^2 with 3 < 9 < 10.
a(624) = 1 since 624 = 2^6 + 4^2 + 12^2 + 20^2 with 4 < 12 < 20.
a(2672) = 1 since 2672 = 2^6 + 4^2 + 36^2 + 36^2 with 4 < 36 = 36.
a(3760) = 1 since 3760 = 0^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.
a(3824) = 1 since 3824 = 2^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.
a(6512) = 1 since 6512 = 2^6 + 12^2 + 52^2 + 60^2 with 12 < 52 < 60.
a(6896) = 1 since 6896 = 2^6 + 36^2 + 44^2 + 60^2 with 36 < 44 < 60.
		

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.NT], 2016-2017.
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Crossrefs

Cf. A000118, A000290, A001014, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273432, A273568.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^6-y^2-z^2],r=r+1],{x,0,n^(1/6)},{y,0,Sqrt[(n-x^6)/3]},{z,y,Sqrt[(n-x^6-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]