A352259 Number of ways to write n as w^6 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

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

Offset: 0

Author

Zhi-Wei Sun, Mar 10 2022

Keywords

• nonn

Comments

Conjecture 1: (i) a(n) > 0 for every n = 0,1,2,.... Moreover, 106, 744, 5469 and 331269 are the only nonnegative integers not in the set {w + x^2 + 2*y^2 + 3*z^2 + x*y*z: w = 0,1; x,y,z = 0,1,2,...}.

(ii) Let k be one of 4, 5, 6, 7. Then each n = 0,1,2,... can be written as 10*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

(iii) Let c be among 1, 3, 4, 6, 7, and let k be 4 or 5. Then every n = 0,1,2,... can be written as c*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

(iv) Each n = 0,1,2,... can be written as 9*w^4 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

Conjecture 2: Every n = 0,1,2,... can be written as 2*w^4 + 3*x^2 + y^2 + z^2 + x*y*z, where w,x,y,z are nonnegative integers.

We have verified Conjectures 1 and 2 for all n <= 10^5.

Examples

			a(24) = 1 with 24 = 0^6 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(106) = 1 with 106 = 2^6 + 1^2 + 2*2^2 + 3*3^2 + 1*2*3.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000290, A000583, A000584, A001014, A351723, A351617, A351902, A352286.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[4(n-w^6-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,n^(1/6)},{z,0,Sqrt[(n-w^6)/3]},{y,0,Sqrt[(n-w^6-3z^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]