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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified January 31 18:34 EST 2023. Contains 359980 sequences. (Running on oeis4.)