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%I #18 May 10 2016 09:14:16
%S 728,756,2457,5824,6048,9288,14364,15561,19656,19684,20412,25327,
%T 25389,39816,46592,48384,66339,70336,74304,76167,76895,82251,91000,
%U 94500,114912,119574,124488,150444,152208,157248,157472,163296,202616,203112,240002,248976
%N Numbers n with property that n and 2n are sums of two distinct positive cubes.
%C Both n and 2n may be represented as a sum of two distinct positive cubes in at least one way, e.g., 19656=18^3+24^3, 2*19656=39312=2^3+34^3=15^3+33^3.
%C Which means, both n and 2n are terms in A024670 (numbers that are sums of 2 distinct positive cubes). E.g., n=728=A024670(27) and 2n=1456=A024670(48).
%C 6742008 is the first n such that n and 2*n are both taxi-cab numbers (A001235). The next one is 53936064. In this sequence, there are infinitely many members such that n and 2*n are both taxi-cab numbers. - _Altug Alkan_, May 10 2016
%H Chai Wah Wu, <a href="/A191345/b191345.txt">Table of n, a(n) for n = 1..10000</a>
%t tdpcQ[n_]:=Module[{pr1=PowersRepresentations[n,2,3],pr2= PowersRepresentations[ 2n,2,3]},Length[pr1]>0&& Length[pr2]>0 && !MemberQ[Flatten[pr1],0] &&!MemberQ[Flatten[pr2],0]]; Select[Range[ 250000],tdpcQ] (* _Harvey P. Dale_, Jul 11 2014 *)
%Y Cf. A024670.
%K nonn
%O 1,1
%A _Zak Seidov_, May 30 2011
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