A018888 Numbers which are not the sum of seven nonnegative cubes.

15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454

Offset: 1

Author

Jud McCranie

Keywords

• fini
• full
• nonn

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

Examples

			239 = 1^3 + 4(2^3) + 3(3^3) + 5^3 - requires 9 cubes.
		

References

• 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
• Index entries for sequences related to sums of cubes

Crossrefs

Cf. A018889.

Programs

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 *)

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

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