A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.

49, 81, 121, 125, 169, 196, 216, 225, 243, 289, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2401, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125, 3136

Offset: 1

Author

Altug Alkan, Jan 23 2016

Keywords

• nonn

Comments

Intersection of A001597 and A004432.

Note that this sequence is not the complement of A267321. This sequence is a subsequence for complement of A267321.

Sequence focuses on the equation m^k = x^2 + y^2 + z^2 where x > y > z > 0 and m > 0, k >= 2.

Corresponding exponents are 2, 4, 2, 3, 2, 2, 3, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 4, 2, 2, ...

Examples

			49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.
81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.
121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.
		

Crossrefs

Cf. A001597, A004432, A266927, A267321.

Programs

Mathematica

fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1800, fQ@ # && Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Jan 24 2016, after Ant King at A001597 *)

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, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));