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

1, 6, 8, 27, 42, 48, 64, 78, 125, 147, 162, 196, 216, 336, 343, 384, 456, 512, 624, 722, 729, 750, 1000, 1050, 1134, 1176, 1296, 1331, 1342, 1568, 1573, 1674, 1694, 1728, 2028, 2058, 2106, 2197, 2366, 2387, 2450, 2522, 2646, 2688, 2744, 2899, 3072, 3087, 3211, 3375, 3648, 3698

Offset: 1

Author

Altug Alkan, Jan 15 2016

Keywords

• nonn
• easy

Comments

Motivation was the simple question: What are the squares that are the averages of two positive cubes?

Corresponding squares are 1, 36, 64, 729, 1764, 2304, 4096, 6084, 15625, 21609, 26244, 38416, 46656, 112896, 117649, 147456, 207936, 262144, 389376, 521284, ...

Examples

			42 is a term because 42^2 = (11^3 + 13^3) / 2.
78 is a term because 78^2 = (1^3 + 23^3) / 2.
147 is a term because 147^2 = (7^3 + 35^3) / 2.
1573 is a term because 1573^2 = (77^3 + 165^3) / 2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A003325, A186885.

Programs

Mathematica

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 *)
(* or, much faster: *) Select[Range@ 1000, {} != PowersRepresentations[#^2 2, 2, 3] &] (* Giovanni Resta, Nov 26 2018 *)

PARI

T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=1, 1e4, if(is(2*n^2), print1(n, ", ")));