cp's OEIS Frontend

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

%I A267986 #14 Jan 30 2016 04:29:14
%S A267986 49,81,121,125,169,196,216,225,243,289,324,361,441,484,529,625,676,
%T A267986 729,784,841,900,961,1000,1089,1156,1225,1296,1331,1369,1444,1521,
%U A267986 1681,1764,1849,1936,2025,2116,2187,2197,2209,2401,2500,2601,2704,2744,2809,2916,3025,3125,3136
%N A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.
%C A267986 Intersection of A001597 and A004432.
%C A267986 Note that this sequence is not the complement of A267321. This sequence is a subsequence for complement of A267321.
%C A267986 Sequence focuses on the equation m^k = x^2 + y^2 + z^2 where x > y > z > 0 and m > 0, k >= 2.
%C A267986 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 A267986 49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.
%e A267986 81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.
%e A267986 121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.
%t A267986 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 A267986 (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 A267986 for(n=1, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));
%Y A267986 Cf. A001597, A004432, A266927, A267321.
%K A267986 nonn
%O A267986 1,1
%A A267986 _Altug Alkan_, Jan 23 2016