OFFSET
1,1
COMMENTS
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.
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
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 306.
J. Roberts, Lure of the Integers, entry 239.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.
LINKS
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, Diophantine Equation--3rd Powers
Eric Weisstein's World of Mathematics, Waring's Problem
EXAMPLE
239 = 1^3 + 4(2^3) + 3(3^3) + 5^3 - requires 9 cubes.
MAPLE
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
MATHEMATICA
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 *)
Select[Range[500], PowersRepresentations[#, 7, 3] == {} &] (* Eric W. Weisstein, Sep 18 2024 *)
PROG
(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
CROSSREFS
KEYWORD
fini,full,nonn
AUTHOR
EXTENSIONS
Corrected by T. D. Noe, Sep 05 2006
Corrected the definition. - N. J. A. Sloane, Sep 25 2011
New name from Charles R Greathouse IV, Dec 30 2014
STATUS
approved