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%I #99 Jul 31 2023 16:10:08
%S 0,1,2,3,6,7,8,9,10,11,12,15,16,17,18,19,20,21,24,25,26,27,28,29,30,
%T 33,34,35,36,37,38,39,42,43,44,45,46,47,48,51,52,53,54,55,56,57,60,61,
%U 62,63,64,65,66,69,70,71,72,73,74,75,78,79,80,81,82,83,84,87,88,89,90,91
%N Numbers that are not congruent to 4 or 5 mod 9.
%C Conjecture: n is a sum of three cubes iff n is in this sequence.
%C As of their 2009 paper, Elsenhans and Jahnel did not know of a sum of three cubes that gives 33 or 42.
%C The problem with 33 is cracked, see links below: 8866128975287528^3 + (-8778405442862239)^3 + (-2736111468807040)^3 = 33. - _Alois P. Heinz_, Mar 11 2019
%C Numbers that are congruent to {0, 1, 2, 3, 6, 7, 8} mod 9. - _Wesley Ivan Hurt_, Jul 21 2016
%C Heath-Brown conjectures that n is a sum of three cubes in infinitely many ways iff n is in this sequence (and not at all otherwise). See his paper for a conjectural asymptotic. - _Charles R Greathouse IV_, Mar 12 2019
%C The problem with 42 is cracked by Andrew Booker from University of Bristol and Andrew Sutherland from Massachusetts Institute of Technology, see the link below: 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3. - _Jianing Song_, Sep 07 2019
%C A third solution to writing 3 as a sum of three third powers was found by the same team using 4 million computer-hours. 3 = 569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3. - _Peter Luschny_, Sep 20 2019
%D R. K. Guy, Unsolved Problems in Number Theory, Section D5.
%D Cohen H. 2007. Number Theory Volume I: Tools and Diophantine Equations. Springer Verlag p. 380. - _Artur Jasinski_, Apr 30 2010
%H Harry J. Smith, <a href="/A060464/b060464.txt">Table of n, a(n) for n = 1..2000</a>
%H Nikos Bagis, <a href="https://arxiv.org/abs/2009.11972">On the numbers that are sums of three cubes</a>, arXiv:2009.11972 [math.GM], 2020.
%H Andrew R. Booker, <a href="https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf">Cracking the problem with 33</a>, March 2019
%H Andrew R. Booker and Brady Haran, <a href="https://www.youtube.com/watch?v=ASoz_NuIvP0">42 is the new 33</a>, Numberphile video (2019)
%H Andrew R. Booker and Brady Haran, <a href="https://www.youtube.com/watch?v=zyG8Vlw5aAw">NEWS: The Mystery of 42 is Solved</a>, Numberphile video (2019)
%H Tim Browning and Brady Haran, <a href="https://www.youtube.com/watch?v=wymmCdLdPvM">The Uncracked Problem with 33</a>, Numberphile video (2015)
%H Tim Browning and Brady Haran, <a href="https://www.youtube.com/watch?v=_-M_3oV75Lw">74 is cracked</a>, Numberphile video (2016)
%H Jean-Louis Colliot-Thélène and Olivier Wittenberg, <a href="http://www.math.ens.fr/~wittenberg/troiscubes.pdf">Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines</a>, Amer. J. Math. 134:5 (2012), pp. 1303-1327.
%H Andreas-Stephan Elsenhans and Jörg Jahnel, <a href="http://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt">List of solutions of x^3 + y^3 + z^3 = n for n < 1000 neither a cube nor twice a cube</a>
%H A.-S. Elsenhans, J. Jahnel, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02168-6">New sums of three cubes</a>, Math. Comp. 78 (2009) 1227-1230.
%H Brady Haran, <a href="https://www.youtube.com/watch?v=vv0bHK44Q1s">569936821221962380720</a>, Numberphile video (2020)
%H D. R. Heath-Brown, <a href="https://doi.org/10.1090/S0025-5718-1992-1146835-5">The density of zeros of forms for which weak approximation fails</a>, Mathematics of Computation 59 (1992), pp. 613-623.
%H Sander G. Huisman, <a href="https://arxiv.org/abs/1604.07746">Newer sums of three cubes</a>, arXiv:1604.07746 [math.NT], 2016.
%H H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm">About n=x^3+y^3+z^3</a>
%H Andrew Sutherland, <a href="https://drive.google.com/file/d/1qzD__dviONTqHQH7DBFmsQ0MdCa7ePRg/view">Sums of three cubes</a>, Slides of a talk given May 07 2020 on the Number Theory Web.
%H University of Bristol, <a href="http://www.bristol.ac.uk/news/2019/september/sum-of-three-cubes-.html">Sum of three cubes for 42 finally solved - using real life planetary computer</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Manin_obstruction">Manin obstruction</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F G.f.: x^2*(x^3+x^2+1)*(x^3+x+1) / ( (1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, Jul 21 2016: (Start)
%F a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; a(n) = a(n-7) + 9 for n>7.
%F a(n) = (63*n - 63 + 2*(n mod 7) + 2*((n+1) mod 7) - 12*((n+2) mod 7) + 2*((n+3) mod 7) + 2*((n+4) mod 7) + 2*((n+5) mod 7) + 2*((n+6) mod 7))/49.
%F a(7k) = 9k-1, a(7k-1) = 9k-2, a(7k-2) = 9k-3, a(7k-3) = 9k-6, a(7k-4) = 9k-7, a(7k-5) = 9k-8, a(7k-6) = 9k-9. (End)
%e 30 belongs to this sequence because it has the partition as sum of 3 cubes 30 = (-283059965)^3 + (-2218888517)^3 + (2220422932)^3. - _Artur Jasinski_, Apr 30 2010, edited by _M. F. Hasler_, Nov 10 2015
%p for n from 0 to 100 do if n mod 9 <> 4 and n mod 9 <> 5 then printf(`%d,`, n) fi:od:
%t a = {}; Do[If[(Mod[n, 9] == 4) || (Mod[n, 9] == 5), , AppendTo[a, n]], {n, 1, 300}]; a (* _Artur Jasinski_, Apr 30 2010 *)
%t Which[Mod[#,9]==4,Nothing,Mod[#,9]==5,Nothing,True,#]&/@Range[0,100] (* _Harvey P. Dale_, Jul 31 2023 *)
%o (PARI) n=-1; for (m=0, 4000, if (m%9!=4 && m%9!=5, write("b060464.txt", n++, " ", m)); if (n==2000, break)) \\ _Harry J. Smith_, Jul 05 2009
%o (PARI) concat(0, Vec(x^2*(x^3+x^2+1)*(x^3+x+1)/((1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2) + O(x^100))) \\ _Altug Alkan_, Nov 06 2015
%o (PARI) a(n)=n\7*9+[0, 1, 2, 3, 6, 7, 8][n%7+1] \\ _Charles R Greathouse IV_, Nov 06 2015
%o (Magma) [n : n in [0..150] | n mod 9 in [0, 1, 2, 3, 6, 7, 8]]; // _Wesley Ivan Hurt_, Jul 21 2016
%o (GAP) A060464:=Filtered([0..100],n->n mod 9 <>4 and n mod 9 <>5); # _Muniru A Asiru_, Feb 17 2018
%Y Cf. A060465, A060466, A060467, A334521, A334522.
%Y A156638 is the complement of this sequence.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Apr 10 2001
%E More terms from _James A. Sellers_, Apr 11 2001
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