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%I #131 Oct 16 2023 23:39:29
%S 1729,4104,13832,20683,32832,39312,40033,46683,64232,65728,110656,
%T 110808,134379,149389,165464,171288,195841,216027,216125,262656,
%U 314496,320264,327763,373464,402597,439101,443889,513000,513856,515375,525824,558441,593047,684019,704977
%N Taxi-cab numbers: sums of 2 cubes in more than 1 way.
%C From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"
%C A011541 gives another version of "taxicab numbers".
%C If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - _Altug Alkan_, May 09 2016
%D R. K. Guy, Unsolved Problems in Number Theory, Section D1.
%D G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
%D Ya. I. Perelman, Algebra can be fun, pp. 142-143.
%D H. W. Richmond, On integers which satisfy the equation t^3 +- x^3 +- y^3 +- z^3, Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.
%H Shahar Amitai, <a href="/A001235/b001235.txt">Table of n, a(n) for n = 1..30000</a> (terms a(1)-a(4724) from T. D. Noe, terms a(4725)-a(10000) from Zak Seidov).
%H Shahar Amitai, <a href="/A001235/a001235_1.txt">Python code to generate all taxicab numbers up to N.</a>
%H J. Charles-É, Recreomath, <a href="http://translate.google.com/translate?hl=en&sl=fr&u=http://www.recreomath.qc.ca/dict_ramanujan_n.htm">Ramanujan's Number</a>.
%H A. Grinstein, <a href="https://web.archive.org/web/20040320144821/http://zadok.org/mattandloraine/1729.html">Ramanujan and 1729</a>, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
%H Istanbul Bilgi University, <a href="https://web.archive.org/web/20110712051436/http://cs.bilgi.edu.tr/pages/curiosity_corner/challenges/ramanujans_number.html">Ramanujan and Hardy's Taxi</a>
%H Christopher Lane, The First ten Ta(2) and their double distinct cubic sums representations, <a href="https://web.archive.org/web/20150911221447/http://cdl.best.vwh.net/JavaScript/Cubes.html">Find Ramanujan's Taxi Number using JavaScript</a>. [WayBack Machine copy]
%H J. Leech, <a href="http://dx.doi.org/10.1017/S0305004100032850">Some solutions of Diophantine equations</a>, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
%H J. Loy, <a href="https://web.archive.org/web/20130823014303/http://www.jimloy.com/number/hardy.htm">The Hardy-Ramanujan Number</a>.
%H Mia Muessig, <a href="https://github.com/PhoenixSmaug/taxicab-numbers">Julia code for finding general taxicab numbers</a>
%H Ken Ono and Sarah Trebat-Leder, <a href="http://arxiv.org/abs/1510.00735">The 1729 K3 surface</a>, arXiv:1510.00735 [math.NT], 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation 3rd Powers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TaxicabNumber.html">Taxicab Number</a>
%H D. W. Wilson, <a href="https://cs.uwaterloo.ca/journals/JIS/wilson10.html">The Fifth Taxicab Number is 48988659276962496</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.9.
%e 4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.
%t Select[Range[750000],Length[PowersRepresentations[#,2,3]]>1&] (* _Harvey P. Dale_, Nov 25 2014, with correction by _Zak Seidov_, Jul 13 2015 *)
%o (PARI) is(n)=my(t);for(k=ceil((n/2)^(1/3)),(n-.4)^(1/3),if(ispower(n-k^3,3),if(t,return(1),t=1)));0 \\ _Charles R Greathouse IV_, Jul 15 2011
%o (PARI) T=thueinit(x^3+1,1);
%o is(n)=my(v=thue(T,n)); sum(i=1,#v,v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ _Charles R Greathouse IV_, May 09 2016
%Y Subsequence of A003325.
%Y Cf. A007692, A008917, A011541, A018786, A018850 (primitive solutions), A051347 (allows negatives), A343708, A360619.
%Y Solutions in greater numbers of ways:
%Y (>2): A018787 (A003825 for primitive, A023050 for coprime),
%Y (>3): A023051 (A003826 for primitive),
%Y (>4): A051167 (A155057 for primitive).
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
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