A351723 Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.

0, 1, 2, 4, 5, 8, 9, 10, 13, 14, 16, 17, 18, 20, 22, 25, 26, 28, 29, 32, 34, 36, 37, 38, 40, 41, 44, 45, 49, 50, 52, 53, 54, 58, 61, 62, 64, 65, 68, 70, 72, 73, 74, 76, 77, 80, 81, 82, 85, 88, 89, 90, 92, 94, 97, 98, 100, 101, 104, 106, 108, 109, 110, 112, 113, 116, 117, 118, 121, 122, 125, 128, 130, 133, 134, 136

Offset: 1

Author

Zhi-Wei Sun, Feb 17 2022

Keywords

• nonn

Comments

It is easy to see that no term is congruent to 3 modulo 4.

Conjecture 1: a(n) < 2*n for all n > 0, and a(n)/n has a limit as n tends to the infinity. Also, a(n) <= a(n-1) + a(n-2) for all n > 4.

Conjecture 2: Let S = {x^2 + y^2 + z^2 + x*y*z: x,y,z = 0,1,2,...}.

(i) 7 and 487 are the only nonnegative integers which cannot be written as w^2 + s, where w is a nonnegative integer and s is an element of S. Also, 7, 87 and 267 are the only nonnegative integers which cannot be written as w^3 + s, where w is a nonnegative integer and s is an element of S.

(ii) Let k be 2 or 3. Then each nonnegative integer not congruent to 3 modulo 4 can be written as 4*w^k + s, where w is a nonnegative integer and s is an element of S.

This has been verified for nonnegative integers up to 10^6.

Examples

			a(3) = 2 with 2 = 1^2 + 1^2 + 0^2 + 1*1*0.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Can each natural number be represented by 2*w^2+x^2+y^2+z^2+xyz with x,y,z in {0,1,2,...}? Question 416344 at MathOverflow, Feb. 17, 2022.

Crossrefs

Cf. A000290, A000378, A000578, A351902.

Programs

Mathematica

tab={};Do[n=x^2+y^2+z^2+x*y*z;If[n<=140,tab=Append[tab,n]],{x,0,20},{y,0,x},{z,0,y}];Print[Sort[DeleteDuplicates[tab]]]