A018889 -id:A018889 - cp's OEIS frontend

A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

0, 3, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Martin Renner, Jan 28 2011

Also continued fraction expansion of (9+sqrt(229))/74. - Bruno Berselli, Sep 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A018889, A010854.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181402.

PadRight[{0, 3}, 100, 15] (* Paolo Xausa, May 24 2024 *)

a(n)=if(n>2,15,3*n-3) \\ Charles R Greathouse IV, Oct 07 2015

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + a(n) + A130130(n) = A002283(n).
a(n) = 15 for n > 2. - Charles R Greathouse IV, Sep 09 2011
G.f.: 3*x^2*(1+4*x)/(1-x). - Bruno Berselli, Sep 09 2011
E.g.f.: 3*(5*(exp(x) - 1 - x) - 2*x^2). - Stefano Spezia, May 21 2024

a(5)-a(7) from Lars Blomberg, May 04 2011

A181405 Total number of n-digit numbers requiring 8 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

0, 3, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Martin Renner, Jan 28 2011

Arthur Wieferich proved that only 15 integers require eight cubes, cf. A018889.

A181354(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + a(n) + A171386(n) = A052268(n)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A018889, A181404.

a(n) = A181404(n) - A181404(n-1).

A004829 Numbers that are the sum of at most 7 positive cubes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

N. J. A. Sloane

McCurley proves that every n > exp(exp(13.97)) is in A003330 and hence in this sequence. Siksek proves that all n > 454 are in this sequence. - Charles R Greathouse IV, Jun 29 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
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, Algebra and Number Theory 10:10 (2016), pp. 2093-2119.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences related to sums of cubes

Complement of A018889; subsequence of A003330.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Cf. A018888.

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

Original entry on oeis.org

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

Offset: 1

Jud McCranie

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

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

J. Roberts, Lure of the Integers, entry 239.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.

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

Cf. A018889.

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

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 *)

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

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

A018890 Numbers whose smallest expression as a sum of positive cubes requires exactly 7 cubes.

Original entry on oeis.org

7, 14, 21, 42, 47, 49, 61, 77, 85, 87, 103, 106, 111, 112, 113, 122, 140, 148, 159, 166, 174, 178, 185, 204, 211, 223, 229, 230, 237, 276, 292, 295, 300, 302, 311, 327, 329, 337, 340, 356, 363, 390, 393, 401, 412, 419, 427, 438, 446, 453, 465, 491, 510, 518, 553, 616

Offset: 1

Anonymous

It is conjectured that a(121)=8042 is the last term - Jud McCranie

An unpublished result of Deshouillers-Hennecart-Landreau, combined with Lemma 3 from Bertault, Ramaré, & Zimmermann implies that if there are any terms beyond a(121) = 8042, they are greater than 1.62 * 10^34. - Charles R Greathouse IV, Jan 23 2014

J. Roberts, Lure of the Integers, entry 239.

T. D. Noe, Table of n, a(n) for n = 1..121
F. Bertault, O. Ramaré, and P. Zimmermann, On sums of seven cubes, Math. Comp. 68 (1999), pp. 1303-1310.
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
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

Cf. A004829, A018888, A018889.

Select[Range[700], (pr = PowersRepresentations[#, 7, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

A046040 Numbers that are the sum of 6 but no fewer positive cubes.

Original entry on oeis.org

6, 13, 20, 34, 39, 41, 46, 48, 53, 58, 60, 69, 76, 79, 84, 86, 95, 98, 102, 104, 105, 110, 117, 121, 123, 124, 132, 139, 147, 151, 158, 165, 170, 173, 177, 184, 196, 202, 203, 210, 215, 221, 222, 228, 235, 236, 242, 247, 249, 263, 265, 268, 273, 275, 284, 287

Offset: 1

Eric W. Weisstein

According to the McCurley article, it is conjectured that there are exactly 3922 terms of which the largest is a(3922) = 1290740.

T. D. Noe, Table of n, a(n) for n=1..3922
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
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

Cf. A000578, A003325, A003072, A003327, A003328, A018890, A018889.

Select[Range[300], (pr = PowersRepresentations[#, 6, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

Corrected by Arlin Anderson (starship1(AT)gmail.com).

A047719 Numbers that are the sum of 8 but no fewer nonzero fourth powers.

Original entry on oeis.org

8, 23, 38, 53, 68, 88, 103, 118, 128, 133, 148, 168, 183, 193, 198, 213, 228, 248, 263, 278, 293, 308, 323, 328, 343, 358, 368, 373, 388, 403, 408, 423, 433, 438, 453, 468, 483, 488, 498, 503, 518, 533, 548, 563, 568, 578, 583, 598, 608, 613, 632, 647, 648

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

David A. Corneth, Table of n, a(n) for n = 1..20516 (terms <= 200000)

Cf. A000583, A002377, A018889, A003342.

upto(n)={my(e=8); my(s=sum(k=1, sqrtint(sqrtint(n)), x^(k^4)) + O(x*x^n)); my(p=s^e, q=(1 + s)^(e-1)); select(k->polcoeff(p,k) && !polcoeff(q,k), [1..n])} \\ Andrew Howroyd, Jul 06 2018

A098821 a(n) = (n-2) * 2^(n-1) + 5.

Original entry on oeis.org

4, 4, 5, 9, 21, 53, 133, 325, 773, 1797, 4101, 9221, 20485, 45061, 98309, 212997, 458757, 983045, 2097157, 4456453, 9437189, 19922949, 41943045, 88080389, 184549381, 385875973, 805306373, 1677721605, 3489660933, 7247757317

Offset: 0

Parthasarathy Nambi, Oct 08 2004

			a(5) = 3*2^4 + 5 = 53.

G. H. Hardy and J. E. Littlewood, "Some problems of partitio numerorum (VI): Further researches in Waring's Problem", Math. Z. vol. 23, 1-37, (1925)
T. D. Wooley, "Large improvements in Waring's Problem", Ann. Math. vol. 135, 131-164 (1992)

Eric Weisstein, Waring's problem
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

Cf. A018889, A002804.

Table[(n - 2)*2^(n - 1) + 5, {n, 0, 30}] (* Stefan Steinerberger, Mar 06 2006 *)
LinearRecurrence[{5,-8,4},{4,4,5},40] (* Harvey P. Dale, Feb 22 2013 *)

a(n)=(n-2)<<(n-1)+5 \\ Charles R Greathouse IV, Jul 23 2015

From Colin Barker, Jan 28 2012: (Start)
G.f.: (4-16*x+17*x^2)/(1-5*x+8*x^2-4*x^3).
a(n)=5*a(n-1)-8*a(n-2)+4*a(n-3). (End)

More terms from Stefan Steinerberger, Mar 06 2006

A098823 a(n) = 16(8prime(n) + 7).

Original entry on oeis.org

368, 496, 752, 1008, 1520, 1776, 2288, 2544, 3056, 3824, 4080, 4848, 5360, 5616, 6128, 6896, 7664, 7920, 8688, 9200, 9456, 10224, 10736, 11504, 12528, 13040, 13296, 13808, 14064, 14576, 16368, 16880, 17648, 17904, 19184, 19440, 20208, 20976

Offset: 1

Parthasarathy Nambi, Oct 08 2004

nonn

			4^2 * (8*2 + 7) = 368 when p=2.

Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A018889, A002804.

Table[16*(8*Prime[n] + 7), {n, 1, 40}] (* Stefan Steinerberger, Mar 06 2006 *)

main(m)=forprime(p=2,m,print1(16 * (8 * p + 7),", ")) \\ Anders Hellström, Aug 26 2015

More terms from Stefan Steinerberger, Mar 06 2006

cp's OEIS Frontend

A181404 Total number of positive integers below 10^n requiring 8 positive cubes in their representation as sum of cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A181405 Total number of n-digit numbers requiring 8 positive cubes in their representation as sum of cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A004829 Numbers that are the sum of at most 7 positive cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A018890 Numbers whose smallest expression as a sum of positive cubes requires exactly 7 cubes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A046040 Numbers that are the sum of 6 but no fewer positive cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A047719 Numbers that are the sum of 8 but no fewer nonzero fourth powers.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A098821 a(n) = (n-2) * 2^(n-1) + 5.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

A098823 a(n) = 16(8prime(n) + 7).