A002376 -id:A002376 - cp's OEIS frontend

A163490 Least number k having n representations as the sum of the minimal number of cubes A002376(k).

1, 157, 221, 626, 894, 1488, 1489, 3020, 1912, 1938, 3685, 3659, 4982, 4369, 5279, 13127, 4882, 5305, 8042, 16116, 16620, 18884, 23604, 22514, 22542, 29094, 31353, 27660, 41388, 38883

Offset: 1

Martin Renner, Jan 22 2011, Jan 25 2011

			a(1) = 1 since 1 = 1^3 (1 way with minimal representation).
a(2) = 157 since 157 = 1^3 + 1^3 + 3^3 + 4^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 5^3 (2 ways with minimal representation).
a(3) = 221 since 221 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 6^3 = 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 4^3 + 5^3 (3 ways with minimal representation).

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

Cf. A002376.

t=Table[r=PowersRepresentations[n,9,3]; Sort[Tally[9-Count[#,0]&/@r]][[1,2]],{n,1000}]; u=Union[t]; c=Complement[Range[Max[u]],u]; If[c=={},mx=u[[-1]],mx=c[[1]]-1]; Flatten[Table[Position[t,n,1,1],{n,mx}]]

a(16)-a(30) from Alois P. Heinz, Feb 10 2011

A002828 Least number of squares that add up to n.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 3, 4, 2, 1, 2, 3, 3, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 2, 3, 1, 2, 3, 4, 2, 2, 3, 3, 3, 2, 3, 4, 3, 1, 2, 3, 2, 2, 3, 4, 3, 3, 2, 3, 4, 2, 3, 4, 1, 2, 3, 3, 2, 3, 3, 4, 2, 2, 2, 3, 3, 3, 3, 4, 2, 1, 2, 3, 3, 2, 3, 4, 3, 2, 2, 3, 4, 3, 3, 4, 3, 2, 2, 3, 1, 2, 3, 4, 2, 3

Offset: 0

N. J. A. Sloane

Lagrange's "Four Squares theorem" states that a(n) <= 4.
It is easy to show that this is also the least number of squares that add up to n^3.
a(n) is the number of iterations in f(...f(f(n))...) to reach 0, where f(n) = A262678(n) = n - A262689(n)^2. Allows computation of this sequence without Lagrange's theorem. - Antti Karttunen, Sep 09 2016
It is also easy to show that a(k^2*n) = a(n) for k > 0: Clearly a(k^2*n) <= a(n) but for all 4 cases of a(n) there is no k which would result in a(k^2*n) < a(n). - Peter Schorn, Sep 06 2021

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).

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe with corrections from Michel Marcus)
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Square Number.

Cf. A000290, A000415, A000419, A002376, A004215, A000378, A001481.

Cf. A010052, A025426, A025427, A053610, A062535, A072401, A229062, A255131, A260728, A260731, A260734, A262678, A262689, A262690, A276573.

a002828 0 = 0  -- confessedly  /= 1, as sum [] == 0
a002828 n | a010052 n == 1 = 1
          | a025426 n > 0 = 2 | a025427 n > 0 = 3 | otherwise = 4
-- Reinhard Zumkeller, Feb 26 2015

with(transforms);
sq:=[seq(n^2, n=1..20)];
LAGRANGE(sq,4,120);
# alternative:
f:= proc(n) local F,x;
   if issqr(n) then return 1 fi;
   if nops(select(t -> t[1] mod 4 = 3 and t[2]::odd, ifactors(n)[2])) = 0 then return 2 fi;
   x:= n/4^floor(padic:-ordp(n,2)/2);
   if x mod 8 = 7 then 4 else 3 fi
end proc:
0, seq(f(n),n=1..200); # Robert Israel, Jun 14 2016
# next Maple program:
b:= proc(n, i) option remember; convert(series(`if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(s-> `if`(s>n, 0, x*b(n-s, i)))(i^2))), x, 5), polynom)
    end:
a:= n-> ldegree(b(n, isqrt(n))):
seq(a(n), n=0..105);  # Alois P. Heinz, Oct 30 2021

SquareCnt[n_] := If[SquaresR[1, n] > 0, 1, If[SquaresR[2, n] > 0, 2, If[SquaresR[3, n] > 0, 3, 4]]]; Table[SquareCnt[n], {n, 150}] (* T. D. Noe, Apr 01 2011 *)
sc[n_]:=Module[{s=SquaresR[Range[4],n]},If[First[s]>0,1,Length[ First[ Split[ s]]]+1]]; Join[{0},Array[sc,110]] (* Harvey P. Dale, May 21 2014 *)

istwo(n:int)=my(f);if(n<3,return(n>=0););f=factor(n>>valuation(n, 2)); for(i=1,#f[,1],if(bitand(f[i,2],1)==1&&bitand(f[i,1],3)==3, return(0)));1
isthree(n:int)=my(tmp=valuation(n,2));bitand(tmp,1)||bitand(n>>tmp,7)!=7
a(n)=if(isthree(n), if(issquare(n), !!n, 3-istwo(n)), 4) \\ Charles R Greathouse IV, Jul 19 2011, revised Mar 17 2022

from sympy import factorint
def A002828(n):
    if n == 0: return 0
    f = factorint(n).items()
    if not any(e&1 for p,e in f): return 1
    if all(p&3<3 or e&1^1 for p,e in f): return 2
    return 3+(((m:=(~n&n-1).bit_length())&1^1)&int((n>>m)&7==7)) # Chai Wah Wu, Aug 01 2023

from sympy.core.power import isqrt
def A002828(n):
    dp = [-1] * (n + 1)
    dp[0] = 0
    for i in range(1, n + 1):
        S = []
        r = isqrt(i)
        for j in range(1, r + 1):
            S.append(1 + dp[i - (j**2)])
        dp[i] = min(S)
    return dp[-1] # Darío Clavijo, Apr 21 2025

;; The first one follows Charles R Greathouse IV's PARI-code above:
(define (A002828 n) (cond ((zero? n) n) ((= 1 (A010052 n)) 1) ((= 1 (A229062 n)) 2) (else (+ 3 (A072401 n)))))
(define (A229062 n) (- 1 (A000035 (A260728 n))))
;; We can also compute this without relying on Lagrange's theorem. The following recursion-formula should be used together with the second Scheme-implementation of A262689 given in the Program section that entry:
(definec (A002828 n) (if (zero? n) n (+ 1 (A002828 (- n (A000290 (A262689 n)))))))
;; Antti Karttunen, Sep 09 2016

From Antti Karttunen, Sep 09 2016: (Start)
a(0) = 0; and for n >= 1, if A010052(n) = 1 [when n is a square], a(n) = 1, otherwise, if A229062(n)=1, then a(n) = 2, otherwise a(n) = 3 + A072401(n). [After Charles R Greathouse IV's PARI program.]
a(0) = 0; for n >= 1, a(n) = 1 + a(n - A262689(n)^2), (see comments).
a(n) = A053610(n) - A062535(n).
(End)

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

A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 42, 18, 2, 1, 30, 110, 92, 24, 2, 1, 42, 240, 340, 162, 30, 2, 1, 56, 462, 1010, 780, 252, 36, 2, 1, 72, 812, 2562, 2970, 1500, 362, 42, 2, 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2, 1, 110, 2070, 11832, 26474, 27174, 14240, 4060, 642, 54, 2, 1, 132, 3080, 22530, 66222, 91112, 65226, 26070, 6040, 812, 60, 2

Offset: 1

Ralf Stephan, Feb 20 2005

T(n,k) is the number of integer sequences of length n+1 with sum zero and sum of absolute values 2k. - R. H. Hardin, Feb 23 2009

			Array begins:
  1,   2,     2,      2,       2,        2,         2,          2, ... A040000;
  1,   6,    12,     18,      24,       30,        36,         42, ... A008458;
  1,  12,    42,     92,     162,      252,       362,        492, ... A005901;
  1,  20,   110,    340,     780,     1500,      2570,       4060, ... A008383;
  1,  30,   240,   1010,    2970,     7002,     14240,      26070, ... A008385;
  1,  42,   462,   2562,    9492,    27174,     65226,     137886, ... A008387;
  1,  56,   812,   5768,   26474,    91112,    256508,     623576, ... A008389;
  1,  72,  1332,  11832,   66222,   271224,    889716,    2476296, ... A008391;
  1,  90,  2070,  22530,  151560,   731502,   2777370,    8809110, ... A008393;
  1, 110,  3080,  40370,  322190,  1815506,   7925720,   28512110, ... A008395;
  1, 132,  4422,  68772,  643632,  4197468,  20934474,   85014204, ... A035837;
  1, 156,  6162, 112268, 1219374,  9129276,  51697802,  235895244, ... A035838;
  1, 182,  8372, 176722, 2206932, 18827718, 120353324,  614266354, ... A035839;
  1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
  ...
Antidiagonals:
  1;
  1,  2;
  1,  6,    2;
  1, 12,   12,    2;
  1, 20,   42,   18,    2;
  1, 30,  110,   92,   24,    2;
  1, 42,  240,  340,  162,   30,    2;
  1, 56,  462, 1010,  780,  252,   36,   2;
  1, 72,  812, 2562, 2970, 1500,  362,  42,  2;
  1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48,  2;

Muniru A Asiru, Table of n, a(n) for n = 1..5050 (antidiagonals 1 to 100, flattened)
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, Preserving Privacy and Fidelity via Ehrhart Theory, July 2017.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.
Columns include A002376, A001621.
Main diagonal is in A103882.
Cf. A047085, A103884, A103903, A103998, A143007.

T:=Flat(List([1..12],n->Concatenation([1],List([1..n-1],k->Sum([1..n],i->Binomial(n-k+1,i)*Binomial(k-1,i-1)*Binomial(n-i,k)))))); # Muniru A Asiru, Oct 14 2018

A103881:= func< n,k | k le 0 select 1 else (&+[Binomial(n-k+1, j)*Binomial(k-1, j-1)*Binomial(n-j, k): j in [1..n-k]]) >;
[A103881(n,k): k in [0..n-1], n in [1..15]]; // G. C. Greubel, Oct 16 2018; May 24 2023

T:=proc(n,k) option remember; local i;
if k=0 then 1 else
add( binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k),i=1..n); fi;
end:
g:=n->[seq(T(n-i,i),i=0..n-1)]:
for n from 1 to 14 do lprint(op(g(n))); od:

T[n_, k_]:= (n+1)*(n+k-1)!*HypergeometricPFQ[{1-k,1-n,-n}, {2,-n-k+1}, 1]/(k!*(n-1)!); T[, 0]=1; Flatten[Table[T[n-k, k], {n,12}, {k,0,n-1}]] (* _Jean-François Alcover, Dec 27 2012 *)

A103881(n,k) = if(k==0, 1, sum(j=1, n-k, binomial(n-k+1, j)*binomial(k-1, j-1)*binomial(n-j, k)));
for(n=1, 15, for(k=0, n-1, print1(A103881(n,k), ", "))) \\ G. C. Greubel, Oct 16 2018; May 24 2023

def A103881(n,k): return 1 if k==0 else (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1).simplify()
flatten([[A103881(n,k) for k in range(n)] for n in range(1,16)]) # G. C. Greubel, May 24 2023

T(n,k) = Sum_{i=1..n} C(n+1, i)*C(k-1, i-1)*C(n-i+k, k), T(n,0)=1.
G.f. of n-th row: (Sum_{i=0..n} C(n, i)^2*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).
T(n, k) = (n+1)*binomial(n+k-1,k)*hypergeometric([-n,1-n,1-k], [2,1-n-k], 1), with T(n, k) = 1 (array).
t(n, k) = (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1), with t(n, 0) = 1 (antidiagonals).
Sum_{k=0..n-1} t(n, k) = A047085(n). (End)
From Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).
(n+1)*T(n+1,k) = (n+1)*T(n+1,k-1) + (2*n+1)*(T(n,k) + T(n,k-1)) - n*(T(n-1,k) - T(n-1,k-1)). (End)

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

A002377 Least number of 4th powers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 1, 2, 3

Offset: 1

N. J. A. Sloane

No terms are greater than 19, see A002804. - Charles R Greathouse IV, Aug 01 2013

Seven values of n need the maximum of 19 fourth powers. These form the arithmetic progression {79, 159, 239, 319, 399, 479, 559} each term being congruent to 79 mod 80. For n < 625 the available fourth powers are congruent to 1 or 16 mod 80, requiring 4*16 + 15*1 to sum to 79. However, 625 = 5^4 is congruent to 65 and 1*65 + 14*1 = 79. So for n > 625 and congruent to 79, only 15 fourth powers are needed to satisfy the mod 80 arithmetic. - Peter Munn, Apr 12 2017

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
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).

T. D. Noe, Table of n, a(n) for n = 1..14000
C. A. Bretschneider, Zerlegung der Zahlen bis 4100 in Biquadrate, J. Reine Angew. Math., 46 (1853), 1-28.
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A002828, A002376, A188462, A374012.

Cf. A046049, A046050, A099591.

Cnt4[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 4]] == 0, k++]; k]; Array[Cnt4, 100] (* T. D. Noe, Apr 01 2011 *)
seq[n_] := Module[{v = Table[0, {n}], s, p}, s = Sum[x^(k^4), {k, 1, n^(1/4)}] + O[x]^(n+1); p=1; For[k=1, k <= 19, k++, p *= s; For[i=1, i <= n, i++, If[v[[i]]==0 && Coefficient[p, x, i] != 0, v[[i]] = k]]]; v];
seq[100] (* Jean-François Alcover, Sep 28 2019, after Andrew Howroyd *)

seq(n)={my(v=vector(n), s=sum(k=1, sqrtint(sqrtint(n)), x^(k^4)) + O(x*x^n), p=1); for(k=1, 19, p*=s; for(i=1, n, if(!v[i] && polcoeff(p,i), v[i]=k))); v} \\ Andrew Howroyd, Jul 06 2018

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002377(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,4,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

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

A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.

Original entry on oeis.org

1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272, 4295398733, 8590581749

Offset: 1

N. J. A. Sloane

g(n) is the smallest number s such that every natural number is the sum of at most s n-th powers of natural numbers.
It is known (Kubina and Wunderlich, 1990) that g(n) = 2^n + floor((3/2)^n) - 2 for all n <= 471600000. This formula is conjectured to be correct for all n (see A174420).
Mahler showed that there are only finitely many n's for which this formula fails. - Tomohiro Yamada, Sep 23 2017
This sequence (which corresponds to Waring's original conjecture) is much easier to compute than A079611, the problem of finding the minimal s = G(n) for almost all (= sufficienly large) integers. See Wikipedia for a one-line proof that this value for g(n), conjectured by J. A. Euler in 1772, is indeed a lower bound; it is known to be tight if 2^n*frac((3/2)^n) + floor((3/2)^n) <= 2^n, and no counterexample to this inequality is known. - M. F. Hasler, Jun 29 2014

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers (Basic Books 1996) 252-257.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 195.
G. H. Hardy, Collected Papers. Vols. 1-, Oxford Univ. Press, 1966-; see vol. 1, p. 668.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
S. Pillai, On Waring's Problem, Journal of Indian Math. Soc., 2 (1936), 16-44
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 138.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 239.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 249-250.
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
Edward Waring, Meditationes algebraicae, Cantabrigiae: typis Academicis excudebat J. Archdeacon, 1770.

T. D. Noe, Table of n, a(n) for n = 1..200
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.
Leonard E. Dickson, The Waring Problem and its generalizations, Bulletin of the AMS, 42 (1936) 833-842.
Jeffrey M. Kubina and Marvin C. Wunderlich, Extending Waring's conjecture to 471,600,000, Math. Comp., 55, no. 192 (1990): 815-820.
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math/0412220 [math.NT], 2004.
Feihu Liu and Guoce Xin, On Frobenius Formulas of Power Sequences, arXiv:2210.02722 [math.CO], 2022. See p. 22.
Kurt Mahler, On the fractional parts of the powers of a rational number (II), Mathematika, 4 (1957) 122-124 Math. Rev. 20:33.
Ramin Takloo-Bighash, What about geometry?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 165-185.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, Waring's Problem.
Wikipedia, Waring's Problem.

Cf. A002376, A002377, A079611, A174406, A174420, A297446 (for info on Mathematica functions).

[2^n+Floor((3/2)^n)-2: n in [1..40]]; // Vincenzo Librandi, Aug 15 2015

Maple
```
A002804 := n->2^n+floor( (3/2)^n ) -2;
```

a[n_] := 2^n + Floor[(3/2)^n] - 2; Array[a, 31] (* Robert G. Wilson v, Oct 29 2013 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; a[n_] := 2^n + x[n] - 2; Array[a, 31]  (* Fred Daniel Kline, Jan 11 2018 *)

a(n)=2^n+(3^n>>n)-2 \\ Charles R Greathouse IV, Feb 01 2013

def A002804(n): return (1<>n)-2 # Chai Wah Wu, Jun 25 2024

[2^n+int((3/2)^n)-2 for n in range(1,34)] # Stefano Spezia, Dec 08 2024

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

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)=
{ /* Return last element <=v in sorted array F[] */
    local(j=1);
    while ( F[j]<=v, j+=1 );
    return( F[j-1] );
}
greedy(n,F)=
{
    local(v=n,ct=0);
    while ( v,  v-=last_leq(v,F); ct+=1; );
    return(ct);
}
vector(min(100,F[#F-1]),n,greedy(n,F)) /* show terms */
/* Joerg Arndt, Apr 08 2011 */

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

A079611 Waring's problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.

Original entry on oeis.org

1, 4, 4, 16, 6, 9, 8, 32, 13, 12, 12, 16, 14, 15, 16, 64, 18, 27, 20, 25

Offset: 1

N. J. A. Sloane, Jan 28 2003

The only certain values are G(1) = 1, G(2) = 4 and G(4) = 16.

See A002804 for the simpler problem of Waring's original conjecture, which does not restrict the bound to "sufficiently large" numbers. - M. F. Hasler, Jun 29 2014

			It is known that every sufficiently large number is the sum of 16 fourth powers, and 16 is the smallest number with this property, so a(4) = G(4) = 16. (The numbers 16^k*31 are not the sum of fewer than 16 fourth powers.)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 395 (shows G(4) >= 16).
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

H. Davenport, On Waring's problem for fourth powers, Annals of Mathematics, 40 (1939), 731-747. (Shows that G(4) <= 16.)
Wikipedia, Waring's Problem.
Trevor D. Wooley, On Waring's problem for intermediate powers, arXiv:1602.03221 [math.NT], 2016.

Cf. A002376, A002377, A002804, A174406.

Entry revised Jun 29 2014

A047702 Numbers that are the sum of 3 but no fewer positive cubes.

Original entry on oeis.org

3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, 80, 81, 92, 99, 118, 127, 129, 134, 136, 141, 153, 155, 160, 179, 190, 192, 197, 218, 225, 232, 244, 251, 253, 258, 270, 277, 281, 288, 307, 314, 342, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

			344 is in A003072, but also in A003325; therefore it is not in here.

C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 352.

T. D. Noe, Table of n, a(n) for n=1..1000
C. G. J. Jacobi, Gesammelte Werke.
Index entries for sequences related to sums of cubes

Cf. A000419, A002376, A003072.

N:= 1000: # to get all terms <= N
G3:= series(add(x^(i^3),i=1..floor(N^(1/3)))^3,x,N+1):
G2:= series(add(x^(i^3),i=0..floor(N^(1/3)))^2,x,N+1):
select(t -> coeff(G3,x,t) > 0 and coeff(G2,x,t) = 0, [$1..N]); # Robert Israel, Dec 12 2016

Select[Range[500], (pr = PowersRepresentations[#, 3, 3]; pr != {} && Count[pr, r_ /; (Times @@ r) == 0] == 0) &][[1 ;; 55]]  (* Jean-François Alcover, Apr 08 2011 *)

The numbers in {A003072 MINUS A000578} MINUS A003325. - R. J. Mathar, Apr 13 2008

A047703 Numbers that are the sum of 4 but no fewer positive cubes.

Original entry on oeis.org

4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 82, 88, 89, 93, 100, 107, 108, 119, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 219, 226, 233, 240, 245, 252, 254, 256, 259, 261, 266, 271, 278

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A004215, A002376, A003327.

Select[Range[300], (pr = PowersRepresentations[#, 4, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

A047704 Numbers that are the sum of 5 but no fewer positive cubes.

Original entry on oeis.org

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 68, 71, 75, 78, 83, 90, 94, 96, 97, 101, 109, 115, 116, 120, 131, 138, 143, 146, 150, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 194, 195, 199, 201, 207, 208, 209, 213, 214, 220, 227, 234, 241, 246, 248

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

This sequence is conjectured to have a finite number of terms. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A003328, A002376.

Select[Range[300], (pr = PowersRepresentations[#, 5, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

cp's OEIS Frontend

A163490 Least number k having n representations as the sum of the minimal number of cubes A002376(k).

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A002828 Least number of squares that add up to n.

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A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

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A002377 Least number of 4th powers needed to represent n.

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A002804 (Presumed) solution to Waring's problem: g(n) = 2^n + floor((3/2)^n) - 2.

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

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