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

1,1

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, ...

Links

Table of n, a(n) for n=1..50.

Example

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.

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 *)

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

Crossrefs

Cf. A001597, A004432, A266927, A267321.

Sequence in context: A036307 A294028 A056938 * A337759 A207638 A286095

Adjacent sequences: A267983 A267984 A267985 * A267987 A267988 A267989

Keyword

nonn

Author

Altug Alkan, Jan 23 2016

Status

approved