A282933 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that 8*y^2 - 8*y*z + 9*z^2 is a square.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 25 2017

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*m (k = 0,1,2,... and m = 1, 8, 15, 23, 39, 47, 71, 93, 239, 287, 311, 319, 383, 391, 591, 632, 1663, 2639, 5591, 6236).

(ii) Each n = 0,1,2,... can be written as x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*y^2 - b*y*z + c*z^2 is a square, whenever (a,b,c) is among the ordered triples (6,21,19), (15,33,22), (16,54,39),(18,51,34), (19,53,34), (21,42,22), (22,69,51).

By the linked JNT paper, each n = 0,1,2,... is the sum of a fourth power and three squares, and we can also write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y*(y-z) = 0. Whether y = 0 or y = z, the number 8*y^2 - 8*y*z + 9*z^2 is definitely a square.

First occurrence of k: 1, 2, 5, 6, 10, 18, 26, 25, 41, 85, 81, 101, 105, 90, 201, 146, 321, 341, 261, 325, 297, 370, 585, 306, 906, ..., . Robert G. Wilson v, Feb 25 2017

Examples

			a(8) = 1 since 8 = 0^4 + 0^2 + 2^2 + 2^2 with 8*0^2 - 8*0*2 + 9*2^2 = 6^2.
a(15) = 1 since 15 = 1^4 + 2^2 + 1^2 + 3^2 with 8*2^2 - 8*2*1 + 9*1^2 = 5^2.
a(23) = 1 since 23 = 1^4 + 3^2 + 3^2 + 2^2 with 8*3^2 - 8*3*3 + 9*3^2 = 9^2.
a(591) = 1 since 591 = 3^4 + 5^2 + 1^2 + 22^2 with 8*5^2 - 8*5*1 + 9*1^2 = 13^2.
a(632) = 1 since 632 = 4^4 + 12^2 + 6^2 + 14^2 with 8*12^2 - 8*12*6 + 9*6^2 = 30^2.
a(1663) = 1 since 1663 = 3^4 + 27^2 + 23^2 + 18^2 with 8*27^2 - 8*27*23 + 9*23^2 = 75^2.
a(2639) = 1 since 2639 = 7^4 + 15^2 + 3^2 + 2^2 with 8*15^2 - 8*15*3 + 9*3^2 = 39^2.
a(5591) = 1 since 5591 = 5^4 + 6^2 + 21^2 + 67^2 with 8*6^2 - 8*6*21 + 9*21^2 = 57^2.
a(6236) = 1 since 6236 = 1^4 + 45^2 + 31^2 + 57^2 with 8*45^2 - 8*45*31 + 9*31^2 = 117^2.
		

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, arXiv:1701.05868 [math.NT], 2017.

Crossrefs

Cf. A000118, A000290, A000583, A270969, A271518, A281976, A281977, A282013, A282014, A282494, A282495.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[8y^2-8*y*z+9z^2],r=r+1],{x,0,(n-1)^(1/4)},{y,0,Sqrt[n-1-x^4]},{z,0,Sqrt[n-1-x^4-y^2]}];Print[n," ",r];Continue,{n,1,80}]