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A267415 Integers n such that n^n = (x^3 + y^3) / 2 where x, y > 0, is soluble.

%I A267415 #23 Sep 19 2024 14:25:53
%S A267415 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 A267415 66,69,72,75,76,78,81,84,87,90,93,96,99,102,105,108,111,112,114,117,
%U A267415 120,123,125,126,129,132,135,138,141,144,147,150,153,156,159,162
%N A267415 Integers n such that n^n = (x^3 + y^3) / 2 where x, y > 0, is soluble.
%C A267415 Is there a solution n such that n^n = (x^3 + y^3) / 2 where x > y > 0?
%C A267415 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 A267415 Chai Wah Wu, <a href="/A267415/b267415.txt">Table of n, a(n) for n = 1..228</a>
%e A267415 1 is a term because 1^1 = 1 = (1^3 + 1^3) / 2.
%e A267415 3 is a term because 3^3 = 27 = (3^3 + 3^3) / 2.
%e A267415 8 is a term because 8^8 = 2^24 = (256^3 + 256^3) / 2.
%t A267415 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 A267415 (PARI) T=thueinit('z^3+1);
%o A267415 is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
%o A267415 for(n=0, 28, if(is(2*n^n), print1(n, ", ")));
%Y A267415 Cf. A000312, A003325.
%K A267415 nonn
%O A267415 1,3
%A A267415 _Altug Alkan_, Jan 14 2016
%E A267415 a(13) from _Michael De Vlieger_, Jan 15 2016
%E A267415 a(14)-a(60) from _Chai Wah Wu_, Jan 18 2016