A268307 - OEIS

A268307

Integers n such that A005897(n) is the sum of two positive cubes.

5, 11, 17, 28, 37, 81, 87, 107, 141, 178, 200, 205, 229, 296, 301, 377, 385, 395, 427, 497, 511, 595, 613, 641, 660, 907, 914, 921, 955, 975, 983, 991, 1043, 1129, 1265, 1343, 1369, 1382, 1409, 1537, 1552, 1601, 1819, 1838, 1839, 1917, 1922, 1979, 2205, 2299, 2381, 2581, 2649, 2663

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OFFSET

1,1

COMMENTS

The number of unit cubes on the surface of an n X n X n cube is given by sequence A005897.

With a pair of scales, one might incorrectly think a single cube could be equal to the sum of two cubes. However, we know this is impossible because of Fermat's Last Theorem.

But we can put a 6 X 6 X 6 cube containing only surface unit cubes on one scale: there are 152 unit cubes. In other side of the scale we can put a 3 X 3 X 3 cube and a 5 X 5 X 5 cube, so there are 27 unit cubes and 125 unit cubes, and the two pans balance.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

5 is a term because A005897(5) = 152 = 3^3 + 5^3.

11 is a term because A005897(11) = 728 = 6^3 + 8^3.

17 is a term because A005897(17) = 1736 = 2^3 + 12^3.

MATHEMATICA

Select[Range@ 2700, Length[PowersRepresentations[6 #^2 + 2, 2, 3] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Feb 01 2016 *)

PROG

(PARI) T = thueinit('z^3+1);

is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;

a(n) = if(n, 6*n^2+2, 1);