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A267477
Integers n such that n^2 = (x^3 + y^3) / 2 where x, y > 0, is soluble.
3
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
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
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Jan 15 2016
STATUS
approved