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

%I

%S 49,81,121,125,169,196,216,225,243,289,324,361,441,484,529,625,676,

%T 729,784,841,900,961,1000,1089,1156,1225,1296,1331,1369,1444,1521,

%U 1681,1764,1849,1936,2025,2116,2187,2197,2209,2401,2500,2601,2704,2744,2809,2916,3025,3125,3136

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

%C Intersection of A001597 and A004432.

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

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

%C 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, ...

%e 49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.

%e 81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.

%e 121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.

%t 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 *)

%o (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)));

%o for(n=1, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));

%Y Cf. A001597, A004432, A266927, A267321.

%K nonn

%O 1,1

%A _Altug Alkan_, Jan 23 2016

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Last modified September 22 13:53 EDT 2020. Contains 337289 sequences. (Running on oeis4.)