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A018888
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Numbers which are not the sum of seven nonnegative cubes.
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5
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15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454
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OFFSET
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1,1
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COMMENTS
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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.
23 and 239 require 9 cubes and no numbers require > 9 cubes.
Sequence is conjectured to be complete.
Kadiri shows that a(n) < e^71000. - Charles R Greathouse IV, Dec 30 2014
Siksek shows that this sequence is complete. - Charles R Greathouse IV, May 05 2015
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REFERENCES
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J. Roberts, Lure of the Integers, entry 239.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.
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LINKS
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Table of n, a(n) for n=1..17.
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau; appendix by I. Gusti Putu Purnaba, 7373170279850, Math. Comp. 69 (2000), 421-439.
N. D. Elkies, Every even number greater than 454 is the sum of seven cubes, arXiv 1009.3983.
H. Kadiri, Short effective intervals containing primes in arithmetic progressions and the seven cubes problem, Math. Comp. 77 (2008), pp. 1733-1748.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
Samir Siksek, Every integer greater than 454 is the sum of at most seven positive cubes, arXiv:1505.00647 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Waring's Problem
Index entries for sequences related to sums of cubes
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EXAMPLE
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239 = 1^3 + 4(2^3) + 3(3^3) + 5^3 - requires 9 cubes.
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MAPLE
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N:= 10000:
C1:= {seq(i^3, i=0..floor(N^(1/3)))}:
C2:= select(`<=`, {seq(seq(a+b, a=C1), b=C1)}, N):
C3:= select(`<=`, {seq(seq(a+b, a=C1), b=C2)}, N):
C5:= select(`<=`, {seq(seq(a+b, a=C2), b=C3)}, N):
C7:= select(`<=`, {seq(seq(a+b, a=C2), b=C5)}, N):
{$1..N} minus C7; # Robert Israel, Dec 30 2014
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MATHEMATICA
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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
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PROG
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(PARI) S=sum(n=0, 7, x^n^3, O(x^455)); v=Vec(S^7); v=v[2..#v];
for(n=1, #v, if(v[n]==0, print1(n", "))) \\ Charles R Greathouse IV, May 05 2015
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CROSSREFS
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Cf. A018889.
Sequence in context: A066758 A338119 A297931 * A115174 A092783 A108638
Adjacent sequences: A018885 A018886 A018887 * A018889 A018890 A018891
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KEYWORD
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fini,full,nonn
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AUTHOR
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Jud McCranie
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EXTENSIONS
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Corrected by T. D. Noe, Sep 05 2006
Corrected the definition (this question is still open). - N. J. A. Sloane, Sep 25 2011
New name from Charles R Greathouse IV, Dec 30 2014
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STATUS
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approved
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