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A267477
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Integers n such that n^2 = (x^3 + y^3) / 2 where x, y > 0, is soluble.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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, ...
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LINKS
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Giovanni Resta, Table of n, a(n) for n = 1..10000
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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, ", ")));
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CROSSREFS
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Cf. A000290, A003325, A186885.
Sequence in context: A107366 A024873 A066231 * A237290 A229335 A007829
Adjacent sequences: A267474 A267475 A267476 * A267478 A267479 A267480
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KEYWORD
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nonn,easy
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AUTHOR
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Altug Alkan, Jan 15 2016
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STATUS
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approved
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