%I #23 Sep 19 2024 14:25:53
%S 0,1,3,6,8,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,64,
%T 66,69,72,75,76,78,81,84,87,90,93,96,99,102,105,108,111,112,114,117,
%U 120,123,125,126,129,132,135,138,141,144,147,150,153,156,159,162
%N Integers n such that n^n = (x^3 + y^3) / 2 where x, y > 0, is soluble.
%C Is there a solution n such that n^n = (x^3 + y^3) / 2 where x > y > 0?
%C The answer to the above question is yes: 76^76 = (523974089123227128080087214816032969930445946880^3 + 314384453473936276848052328889619781958267568128^3)/2. Other examples include 112^112 and 172^172. - _Chai Wah Wu_, Jan 18 2016
%H Chai Wah Wu, <a href="/A267415/b267415.txt">Table of n, a(n) for n = 1..228</a>
%e 1 is a term because 1^1 = 1 = (1^3 + 1^3) / 2.
%e 3 is a term because 3^3 = 27 = (3^3 + 3^3) / 2.
%e 8 is a term because 8^8 = 2^24 = (256^3 + 256^3) / 2.
%t Select[Range@ 24, Resolve[Exists[{x, y}, And[Reduce[#^# == (x^3 + y^3)/2, {x, y}, Integers], x > 0, y > 0]]] &] (* _Michael De Vlieger_, Jan 15 2016 *)
%o (PARI) T=thueinit('z^3+1);
%o is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
%o for(n=0, 28, if(is(2*n^n), print1(n, ", ")));
%Y Cf. A000312, A003325.
%K nonn
%O 1,3
%A _Altug Alkan_, Jan 14 2016
%E a(13) from _Michael De Vlieger_, Jan 15 2016
%E a(14)-a(60) from _Chai Wah Wu_, Jan 18 2016