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

1,2

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, ...

Links

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

Example

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.

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 *)

Prog

(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, ", ")));

Crossrefs

Cf. A000290, A003325, A186885.

Sequence in context: A107366 A024873 A066231 * A237290 A229335 A007829

Adjacent sequences: A267474 A267475 A267476 * A267478 A267479 A267480

Keyword

nonn,easy

Author

Altug Alkan, Jan 15 2016

Status

approved