A267983 Integers n such that n^3 = (x^2 + y^2 + z^2) / 3 where x > y > z > 0, is soluble.

3, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 94

Offset

1,1

Comments

Motivation was this simple question: What are the cubes that are the averages of 3 nonzero distinct squares?

Corresponding cubes are 27, 216, 343, 729, 1000, 1331, 1728, 2744, 3375, 4913, 5832, 6859, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 27000, ...

Complement of this sequence for positive integers is 1, 2, 4, 5, 8, 13, 16, 20, 21, 29, 32, 37, 45, 52, 53, 61, 64, 69, 77, ...

The positive cubes that are not the averages of 3 nonzero distinct squares are 1, 8, 64, 125, 512, 2197, 4096, 8000, 9261, 24389, 32768, 50653, 91125, ...

Links

Table of n, a(n) for n=1..71.

Example

3 is a term since 3^3 is the average of 1^2, 4^2, 8^2. 3^3 = (1^2 + 4^2 + 8^2) / 3.

Mathematica

Select[Range@ 94, Resolve[Exists[{x, y, z}, Reduce[#^3 == (x^2 + y^2 + z^2)/3, {x, y, z}, Integers], x > y > z > 0]] &] (* Michael De Vlieger, Jan 24 2016 *)

Prog

(PARI) isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e2, if(isA004432(3*n^3), print1(n, ", ")));

Crossrefs

Cf. A004432, A117642.

Sequence in context: A067751 A283762 A088045 * A344153 A334521 A108057

Adjacent sequences: A267980 A267981 A267982 * A267984 A267985 A267986

Keyword

nonn

Author

Altug Alkan, Jan 23 2016

Status

approved