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A351723
Numbers of the form x^2 + y^2 + z^2 + x*y*z with x,y,z nonnegative integers.
6
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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 17 2022
STATUS
approved