cp's OEIS Frontend

A145553 Numbers n such that n^2 can be expressed as the sum of 2 positive cubes in exactly 2 different ways.

%I A145553 #31 May 17 2016 09:13:52
%S A145553 77976,223587,623808,894348,1788696,2105352,2989441,4298427,4672423,
%T A145553 4990464,5986575,6036849,7154784,8437832,9747000,14309568,16842816,
%U A145553 23915528,24147396,24770529,26745768,27948375,34387416,34634719,36570744,37379384,39923712,47892600
%N A145553 Numbers n such that n^2 can be expressed as the sum of 2 positive cubes in exactly 2 different ways.
%C A145553 This is conjectured to be an infinite sequence.
%C A145553 Subsequence of A051302. [_R. J. Mathar_, Oct 14 2008]
%C A145553 First differs from A051302 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
%C A145553 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
%H A145553 Ray Chandler, <a href="/A145553/b145553.txt">Table of n, a(n) for n = 1..771</a>
%e A145553 a(1): 77976^2 = 6080256576 = 1824^3 + 228^3 = 1710^3 + 1026^3;
%e A145553 a(2): 223587^2 = 49991146569 = 3666^3 + 897^3 = 3276^3 + 2457^3;
%e A145553 a(3): 623808^2 = 389136420864 = 7296^3 + 912^3 = 6840^3 + 4104^3;
%e A145553 a(4): 894348^2 = 799858345104 = 9282^3 + 546^3 = 9009^3 + 4095^3.
%Y A145553 Cf. A145552, A051302.
%K A145553 nonn
%O A145553 1,1
%A A145553 Iain Renfrew (iain.renfrew(AT)btinternet.com), Oct 13 2008
%E A145553 a(5)-a(15) from _Zak Seidov_, Oct 15 2008
%E A145553 Extended by _Ray Chandler_, Nov 22 2011