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A267416 Perfect powers of the form x^3 + y^3 where x and y are distinct positive integers.

%I A267416 #25 Oct 04 2017 01:08:11
%S A267416 9,243,576,6561,9604,28224,36864,51984,97344,140625,177147,275625,
%T A267416 345744,419904,450241,614656,717409,1028196,1058841,1399489,1500625,
%U A267416 1590121,1750329,1806336,2359296,3326976,4782969,6230016,7001316,7962624,8340544,9000000
%N A267416 Perfect powers of the form x^3 + y^3 where x and y are distinct positive integers.
%C A267416 Subsequence of A267088.
%C A267416 Intersection of A001597 and A024670.
%C A267416 Obviously, because of Fermat's Last Theorem, a(n) cannot be a cube.
%C A267416 Corresponding exponents are 2, 5, 2, 8, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 4, 2, ...
%C A267416 Motivation for this sequence is the equation m^k = x^3 + y^3 where m > 0, k >= 2 and x, y are distinct positive integers.
%H A267416 Chai Wah Wu, <a href="/A267416/b267416.txt">Table of n, a(n) for n = 1..4707</a>
%e A267416 9 is a term because 9 = 3^2 = 1^3 + 2^3.
%e A267416 243 is a term because 243 = 3^5 = 3^3 + 6^3.
%e A267416 576 is a term because 576 = 24^2 = 4^3 + 8^3.
%e A267416 51984 is a term because 51984 = 228^2 = 11^3 + 37^3.
%t A267416 Union@ Select[Plus @@@ Union@ Map[Sort, Permutations[Range[210]^3, {2}]], # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* _Michael De Vlieger_, Jan 15 2016, after _Ant King_ at A001597 *)
%o A267416 (PARI) is(n) = for( i=ceil(sqrtn(n\2+1, 3)), sqrtn(n-.5, 3), ispower(n-i^3, 3) & return(1));
%o A267416 for(n=1, 1e7, if(is(n) && ispower(n), print1(n, ", ")));
%Y A267416 Cf. A001597, A024670, A267088.
%K A267416 nonn
%O A267416 1,1
%A A267416 _Altug Alkan_, Jan 14 2016