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

1,3

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.

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.

Example

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

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]]]

Crossrefs

Cf. A000290, A000378, A000578, A351902.

Sequence in context: A022549 A045704 A368796 * A169612 A084581 A121996

Adjacent sequences: A351720 A351721 A351722 * A351724 A351725 A351726

Keyword

nonn

Author

Zhi-Wei Sun, Feb 17 2022

Status

approved