cp's OEIS Frontend

A267477 Integers n such that n^2 = (x^3 + y^3) / 2 where x, y > 0, is soluble.

%I A267477 #15 Nov 26 2018 08:49:22
%S A267477 1,6,8,27,42,48,64,78,125,147,162,196,216,336,343,384,456,512,624,722,
%T A267477 729,750,1000,1050,1134,1176,1296,1331,1342,1568,1573,1674,1694,1728,
%U A267477 2028,2058,2106,2197,2366,2387,2450,2522,2646,2688,2744,2899,3072,3087,3211,3375,3648,3698
%N A267477 Integers n such that n^2 = (x^3 + y^3) / 2 where x, y > 0, is soluble.
%C A267477 Motivation was the simple question: What are the squares that are the averages of two positive cubes?
%C A267477 Corresponding squares are 1, 36, 64, 729, 1764, 2304, 4096, 6084, 15625, 21609, 26244, 38416, 46656, 112896, 117649, 147456, 207936, 262144, 389376, 521284, ...
%H A267477 Giovanni Resta, <a href="/A267477/b267477.txt">Table of n, a(n) for n = 1..10000</a>
%e A267477 42 is a term because 42^2 = (11^3 + 13^3) / 2.
%e A267477 78 is a term because 78^2 = (1^3 + 23^3) / 2.
%e A267477 147 is a term because 147^2 = (7^3 + 35^3) / 2.
%e A267477 1573 is a term because 1573^2 = (77^3 + 165^3) / 2.
%t A267477 Select[Range@1000, Resolve@ Exists[{x, y}, And[Reduce[#^2 == (x^3 + y^3)/2, {x, y}, Integers], x > 0, y > 0]] &] (* _Michael De Vlieger_, Jan 16 2016 *)
%t A267477 (* or, much faster: *) Select[Range@ 1000, {} != PowersRepresentations[#^2 2, 2, 3] &] (* _Giovanni Resta_, Nov 26 2018 *)
%o A267477 (PARI) T = thueinit('z^3+1);
%o A267477 is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
%o A267477 for(n=1, 1e4, if(is(2*n^2), print1(n, ", ")));
%Y A267477 Cf. A000290, A003325, A186885.
%K A267477 nonn,easy
%O A267477 1,2
%A A267477 _Altug Alkan_, Jan 15 2016