A018888 -id:A018888 - cp's OEIS frontend

A002376 Least number of positive cubes needed to sum to n.

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 1, 2, 3, 4, 5, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 5, 6, 7, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 3, 4, 5, 6, 7, 6, 7, 4, 4, 5, 2, 3, 4, 5, 6, 5, 5, 6, 3, 4, 5, 6, 7, 6, 6

Offset: 1

N. J. A. Sloane

No terms are greater than 9, see A002804. - Charles R Greathouse IV, Aug 01 2013

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. R. Zornow, De compositione numerorum e cubis integris positivus, J. Reine Angew. Math., 14 (1835), 276-280.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A003325 (numbers requiring 2 cubes), A047702 (numbers requiring 3 cubes), A047703 (numbers requiring 4 cubes), A047704 (numbers requiring 5 cubes), A046040 (numbers requiring 6 cubes), A018890 (numbers requiring 7 cubes), A018888 (numbers requiring 8 or 9 cubes), A055401 (cubes needed by greedy algorithm).

f:= proc(n) option remember;
  min(seq(procname(n - i^3)+1, i=1..floor(n^(1/3))))
end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Jun 30 2017

CubesCnt[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 3]] == 0, k++]; k]; Array[CubesCnt, 100] (* T. D. Noe, Apr 01 2011 *)

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002376(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,3,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

The g.f. conjectured by Simon Plouffe in his 1992 dissertation,

-(-1-z-z^2-z^3-z^4-z^5-z^6+6*z^7)/(z+1)/(z^2+1)/(z^4+1)/(z-1)^2, is incorrect: the first wrong coefficient is that of z^26. - Robert Israel, Jun 30 2017

More terms from Arlin Anderson (starship1(AT)gmail.com)

A055401 Number of positive cubes needed to sum to n using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7

Offset: 0

Henry Bottomley, May 16 2000

Define f(n) = n - k^3 where (k+1)^3 > n >= k^3; a(n) = number of steps such that f(f(...f(n)))= 0.

Also sum of digits when writing n in base where place values are positive cubes, cf. A000433. [Reinhard Zumkeller, May 08 2011]

			a(32)=6 because 32=27+1+1+1+1+1 (not 32=8+8+8+8).
a(33)=7 because 33=27+1+1+1+1+1+1 (not 33=8+8+8+8+1).

Antti Karttunen & Reinhard Zumkeller (terms 1-10000), Table of n, a(n) for n = 0..10000

Cf. A018888, A055400.
Cf. A002376 (least number of positive cubes needed to represent n; differs from this sequence for the first time at n=32, where a(32)=6, while A002376(32)=4).
Cf. A053610, A048766, A000578, A000433.
Cf. also A261225, A261226, A261227, A261228, A261229.

a055401 n = s n $ reverse $ takeWhile (<= n) $ tail a000578_list where
  s _ []                 = 0
  s m (x:xs) | x > m     = s m xs
             | otherwise = m' + s r xs where (m',r) = divMod m x
-- Reinhard Zumkeller, May 08 2011

f:= proc(n,k) local m, j;
if n = 0 then return 0 fi;
for j from k by -1 while j^3 > n do od:
m:= floor(n/j^3);
m + procname(n-m*j^3, j-1);
end proc:
seq(f(n,floor(n^(1/3))),n=0..100); # Robert Israel, Aug 17 2015

a[0] = 0; a[n_] := {n} //. {b___, c_ /; !IntegerQ[c^(1/3)], d___} :> {b, f = Floor[c^(1/3)]^3, c - f, d} // Length; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 17 2015 *)

F=vector(30,n,n^3); /* modify to get other sequences of "greedy representations" */
last_leq(v,F)={local(j=1); while ( F[j]<=v, j+=1 ); F[j-1]} /* Return last element <=v in sorted array F[] */
greedy(n,F)={local(v=n,ct=0); while ( v, v-=last_leq(v,F); ct+=1; ); ct}
vector(min(100,F[#F-1]),n,greedy(n,F)) /* show terms */
/* Joerg Arndt, Apr 08 2011 */

;; With memoization-macro definec.
(definec (A055401 n) (if (zero? n) n (+ 1 (A055401 (A055400 n)))))
;; Antti Karttunen, Aug 16 2015

a(0) = 0; for n >= 1, a(n) = a(n-floor(n^(1/3))^3)+1 = a(A055400(n))+1 = a(n-A048762(n))+1.

a(0) = 0 prepended by Antti Karttunen, Aug 16 2015

A018889 Numbers whose shortest representation as a sum of positive cubes requires exactly 8 cubes.

Original entry on oeis.org

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

Offset: 1

Anonymous

Wieferich proved that 167 is the unique prime in this sequence. - Jonathan Vos Post, Sep 23 2006

Joe Roberts, Lure of the Integers, entry 239.

Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
G. L. Honaker, Jr. and Chris Caldwell, et al., A Prime Curios Page.
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, Warings Problem
Index entries for sequences related to sums of cubes

Subsequence of A018888.

max = 500; nn = Union[(#*#).# & /@ Tuples[Range[0, 7], {7}]][[1 ;; max]]; Select[{#, PowersRepresentations[#, 8, 3]} & /@ Complement[Range[max], nn] , #[[2]] != {} &][[All, 1]] (* Jean-François Alcover, Jul 21 2011 *)

Corrected by Arlin Anderson.
Additional comments from Jud McCranie.
Edited by N. J. A. Sloane, Aug 10 2022

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.

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 *)

A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

Original entry on oeis.org

212, 364, 420, 428

Offset: 1

Jonathan Vos Post, Mar 26 2009

Boklan and Elkies: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

Elkies [2010]: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N. - Jonathan Vos Post, Sep 22 2010

U. V. Linnik: On the representation of large numbers as sums of seven cubes. Rec. Math. [=Mat. Sbornik] N.S. 12(54) (1943), 218-224.
L. E. Dickson: All integers except 23 and 239 are the sums of 8 cubes. Bull. Amer. Math. Soc. 45 (1939), 588-591.

E. Waring: Meditationes Algebraicae [3rd ed. (1782)]: an English translation of the work of Edward Waring, edited and translated from the Latin by Dennis Weeks. Providence, RI: Amer. Math. Soc., 1991. [MR1146921]
Kent D. Boklan and Noam D. Elkies, Every multiple of 4 except 212, 364, 420, and 428 is the sum of seven cubes, arXiv:0903.4503, Mar 26, 2009.
Noam D. Elkies, Every even number greater than 454 is the sum of seven cubes, Sep 21, 2010. [From _Jonathan Vos Post_, Sep 22 2010]

Cf. A000578, A003072, A003325, A003327, A003328, A003329, A057907, A122730.

Subsequence of A018888. - Charles R Greathouse IV, Apr 16 2010

Definition corrected by Jonathan Sondow, Mar 14 2014

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A002376 Least number of positive cubes needed to sum to n.

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A055401 Number of positive cubes needed to sum to n using the greedy algorithm.

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A018889 Numbers whose shortest representation as a sum of positive cubes requires exactly 8 cubes.

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

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

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A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

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