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

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, 66, 69, 72, 75, 76, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 112, 114, 117, 120, 123, 125, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162

Offset: 1

Author

Altug Alkan, Jan 14 2016

Keywords

• nonn

Comments

Is there a solution n such that n^n = (x^3 + y^3) / 2 where x > y > 0?

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

Examples

			1 is a term because 1^1 = 1 = (1^3 + 1^3) / 2.
3 is a term because 3^3 = 27 = (3^3 + 3^3) / 2.
8 is a term because 8^8 = 2^24 = (256^3 + 256^3) / 2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..228

Crossrefs

Cf. A000312, A003325.

Programs

Mathematica

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 *)

PARI

T=thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=0, 28, if(is(2*n^n), print1(n, ", ")));

Extensions

a(13) from Michael De Vlieger, Jan 15 2016

a(14)-a(60) from Chai Wah Wu, Jan 18 2016