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%I #41 Jun 04 2022 22:36:09
%S 15,22,23,50,114,167,175,186,212,231,238,239,303,364,420,428,454
%N Numbers which are not the sum of seven nonnegative cubes.
%C Old name: Write n = m_1^3 + ... +m_k^3 where the m_i are positive integers and k is minimal; sequence gives conjectured list of numbers for which k = 8 or 9.
%C 23 and 239 require 9 cubes and no numbers require > 9 cubes.
%C Sequence is conjectured to be complete.
%C Kadiri shows that a(n) < e^71000. - _Charles R Greathouse IV_, Dec 30 2014
%C Siksek shows that this sequence is complete. - _Charles R Greathouse IV_, May 05 2015
%D J. Roberts, Lure of the Integers, entry 239.
%D F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.
%H Jan Bohman and Carl-Erik Froberg, <a href="http://dx.doi.org/10.1007/BF01934077">Numerical investigation of Waring's problem for cubes</a>, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
%H Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau; appendix by I. Gusti Putu Purnaba, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01116-3">7373170279850</a>, Math. Comp. 69 (2000), 421-439.
%H N. D. Elkies, <a href="http://arxiv.org/abs/1009.3983">Every even number greater than 454 is the sum of seven cubes</a>, arXiv 1009.3983.
%H H. Kadiri, <a href="http://www.cs.uleth.ca/~kadiri/articles/Prime-04-09-07.pdf">Short effective intervals containing primes in arithmetic progressions and the seven cubes problem</a>, Math. Comp. 77 (2008), pp. 1733-1748.
%H K. S. McCurley, <a href="http://dx.doi.org/10.1016/0022-314X(84)90100-8">An effective seven-cube theorem</a>, J. Number Theory, 19 (1984), 176-183.
%H Samir Siksek, <a href="http://arxiv.org/abs/1505.00647">Every integer greater than 454 is the sum of at most seven positive cubes</a>, arXiv:1505.00647 [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/WaringsProblem.html">Waring's Problem</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%e 239 = 1^3 + 4(2^3) + 3(3^3) + 5^3 - requires 9 cubes.
%p N:= 10000:
%p C1:= {seq(i^3, i=0..floor(N^(1/3)))}:
%p C2:= select(`<=`,{seq(seq(a+b,a=C1),b=C1)},N):
%p C3:= select(`<=`,{seq(seq(a+b,a=C1),b=C2)},N):
%p C5:= select(`<=`,{seq(seq(a+b,a=C2),b=C3)},N):
%p C7:= select(`<=`,{seq(seq(a+b,a=C2),b=C5)},N):
%p {$1..N} minus C7; # _Robert Israel_, Dec 30 2014
%t nn=10000; t=CoefficientList[Series[Sum[x^(k^3), {k,0,Floor[nn^(1/3)]}]^7, {x,0,nn}], x]; Flatten[Position[t,0]]-1 (* _T. D. Noe_, Sep 05 2006 *)
%o (PARI) S=sum(n=0,7,x^n^3,O(x^455)); v=Vec(S^7);v=v[2..#v];
%o for(n=1,#v,if(v[n]==0,print1(n", "))) \\ _Charles R Greathouse IV_, May 05 2015
%Y Cf. A018889.
%K fini,full,nonn
%O 1,1
%A _Jud McCranie_
%E Corrected by _T. D. Noe_, Sep 05 2006
%E Corrected the definition (this question is still open). - _N. J. A. Sloane_, Sep 25 2011
%E New name from _Charles R Greathouse IV_, Dec 30 2014
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