

A145553


Numbers n such that n^2 can be expressed as the sum of 2 positive cubes in exactly 2 different ways.


3



77976, 223587, 623808, 894348, 1788696, 2105352, 2989441, 4298427, 4672423, 4990464, 5986575, 6036849, 7154784, 8437832, 9747000, 14309568, 16842816, 23915528, 24147396, 24770529, 26745768, 27948375, 34387416, 34634719, 36570744, 37379384, 39923712, 47892600
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OFFSET

1,1


COMMENTS

This is conjectured to be an infinite sequence.
Subsequence of A051302. [R. J. Mathar, Oct 14 2008]
First differs from A051302 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
If n is a term of this sequence, then n^2 = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs. If n^2 = a^3 + b^3 = c^3 + d^3, then (n*k^3)^2 = n^2*k^6 = k^6*(a^3 + b^3) = k^6*(c^3 + d^3) = (a*k^2)^3 + (b*k^2)^3 = (c*k^2)^3 + (d*k^2)^3. It is obvious that if (a, b) and (c, d) are distinct, then (k^2*a, k^2*b), (k^2*c, k^2*d) are also distinct for all nonzero values of k. So if n is in this sequence and n*k^3 is not in A155961, then n*k^3 is in this sequence for all k > 0. If this sequence is not infinite, then there are infinitely many consecutive k values for any term n such that n*k^3 is in A155961. Is it possible?  Altug Alkan, May 10 2016


LINKS

Ray Chandler, Table of n, a(n) for n = 1..771


EXAMPLE

a(1): 77976^2 = 6080256576 = 1824^3 + 228^3 = 1710^3 + 1026^3;
a(2): 223587^2 = 49991146569 = 3666^3 + 897^3 = 3276^3 + 2457^3;
a(3): 623808^2 = 389136420864 = 7296^3 + 912^3 = 6840^3 + 4104^3;
a(4): 894348^2 = 799858345104 = 9282^3 + 546^3 = 9009^3 + 4095^3.


CROSSREFS

Cf. A145552, A051302.
Sequence in context: A061528 A210141 A051302 * A016819 A016855 A016975
Adjacent sequences: A145550 A145551 A145552 * A145554 A145555 A145556


KEYWORD

nonn


AUTHOR

Iain Renfrew (iain.renfrew(AT)btinternet.com), Oct 13 2008


EXTENSIONS

a(5)a(15) from Zak Seidov, Oct 15 2008
Extended by Ray Chandler, Nov 22 2011


STATUS

approved



