A004999 -id:A004999 - cp's OEIS frontend

A385237 Smallest x such that x^3+y^3 = A004999(n), x and y are nonnegative integers.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 4, 2, 3, 4, 0, 1, 2, 3, 5, 4, 5, 0, 1, 2, 3, 4, 6, 5, 0, 1, 2, 3, 6, 4, 5, 7, 6, 0, 1, 2, 3, 4, 5, 7, 6, 0, 1, 2, 8, 3, 4, 7, 5, 6, 8, 0, 1, 2, 7, 3, 4, 5, 9, 8, 6, 7, 0, 1, 2, 3, 4, 8, 5, 6, 10, 9, 7, 0, 1, 2, 3, 8, 4, 5, 10, 6, 9, 7, 11

Offset: 1

Zhuorui He, Jul 08 2025

nonn

			For n=9, A004999(9) = 35 = 2^3 + 3^3, so a(9) = 2.
For n=17, A004999(17) = 128 = 4^3 + 4^3, so a(17)=4.

Cf. A004999, A229140.

A003325 Numbers that are the sum of 2 positive cubes.

Original entry on oeis.org

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known. [Any further examples are greater than 5 million. - Charles R Greathouse IV, Apr 12 2020] [Any further example is greater than 10^12. - M. F. Hasler, Jan 10 2021]
A113958 is a subsequence; if m is a term then m+k^3 is a term of A003072 for all k > 0. - Reinhard Zumkeller, Jun 03 2006
From James R. Buddenhagen, Oct 16 2008: (Start)
(i) N and N+1 are both the sum of two positive cubes if N=2*(2*n^2 + 4*n + 1)*(4*n^4 + 16*n^3 + 23*n^2 + 14*n + 4), n=1,2,....
(ii) For n >= 2, let N = 16*n^6 - 12*n^4 + 6*n^2 - 2, so N+1 = 16*n^6 - 12*n^4 + 6*n^2 - 1.
Then the identities 16*n^6 - 12*n^4 + 6*n^2 - 2 = (2*n^2 - n - 1)^3 + (2*n^2 + n - 1)^3 16*n^6 - 12*n^4 + 6*n^2 - 1 = (2*n^2)^3 + (2*n^2 - 1)^3 show that N, N+1 are in the sequence. (End)
If n is a term then n*m^3 (m >= 2) is also a term, e.g., 2m^3, 9m^3, 28m^3, and 35m^3 are all terms of the sequence. "Primitive" terms (not of the form n*m^3 with n = some previous term of the sequence and m >= 2) are 2, 9, 28, 35, 65, 91, 126, etc. - Zak Seidov, Oct 12 2011
This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King, May 09 2013
By Fermat's Last Theorem (the special case for exponent 3, proved by Euler, is sufficient), this sequence contains no cubes. - Charles R Greathouse IV, Apr 03 2021

C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
C. G. J. Jacobi, Gesammelte Werke.
Michael Penn, 1674 is not a perfect cube, 2020 video
N. J. A. Sloane, Table of n, a(n) for n = 1..59562
D. Tournes, A Glance on Indian Mathematician Srinivasa Ramanujan(1887-1920). [Text in French]
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Subsequence of A004999 and hence of A045980; supersequence of A202679.

Cf. A024670 (2 distinct cubes), A003072, A001235, A011541, A003826, A010057, A000578, A027750, A010052, A085323 (n such that a(n+1)=a(n)+1).

a003325 n = a003325_list !! (n-1)
a003325_list = filter c2 [1..] where
   c2 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (> 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Mar 24 2012

nn = 2*20^3; Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]] (* T. D. Noe, Oct 12 2011 *)
With[{upto=2000},Select[Total/@Tuples[Range[Ceiling[Surd[upto,3]]]^3,2],#<=upto&]]//Union (* Harvey P. Dale, Jun 11 2016 *)

cubes=sum(n=1, 11, x^(n^3), O(x^1400)); v = select(x->x, Vec(cubes^2), 1); vector(#v, k, v[k]+1) \\ edited by Michel Marcus, May 08 2017

isA003325(n) = for(k=1,sqrtnint(n\2,3), ispower(n-k^3,3) && return(1)) \\ M. F. Hasler, Oct 17 2008, improved upon suggestion of Altug Alkan and Michel Marcus, Feb 16 2016

T=thueinit('z^3+1); is(n)=#select(v->min(v[1],v[2])>0, thue(T,n))>0 \\ Charles R Greathouse IV, Nov 29 2014

list(lim)=my(v=List()); lim\=1; for(x=1,sqrtnint(lim-1,3), my(x3=x^3); for(y=1,min(sqrtnint(lim-x3,3),x), listput(v, x3+y^3))); Set(v) \\ Charles R Greathouse IV, Jan 11 2022

from sympy import integer_nthroot
def aupto(lim):
  cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]
  sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])
  return [s for s in sum_cubes if s <= lim]
print(aupto(1343)) # Michael S. Branicky, Feb 09 2021

Error in formula line corrected by Zak Seidov, Jul 23 2009

A003347 Numbers that are the sum of 2 positive 5th powers.

Original entry on oeis.org

2, 33, 64, 244, 275, 486, 1025, 1056, 1267, 2048, 3126, 3157, 3368, 4149, 6250, 7777, 7808, 8019, 8800, 10901, 15552, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33614, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 65536, 66825, 75856

Offset: 1

N. J. A. Sloane

nonn

			From _David A. Corneth_, Aug 03 2020: (Start)
917552689 is in the sequence as 917552689 = 17^5 + 62^5.
2557575000 is in the sequence as 2557575000 = 45^5 + 75^5.
5828050944 is in the sequence as 5828050944 = 56^5 + 88^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.

Cf. A000584, A003336, A003348, A003358, A088703.

With[{k = 5}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[9], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999  *)

A003358 Numbers that are the sum of 2 nonzero 6th powers.

Original entry on oeis.org

2, 65, 128, 730, 793, 1458, 4097, 4160, 4825, 8192, 15626, 15689, 16354, 19721, 31250, 46657, 46720, 47385, 50752, 62281, 93312, 117650, 117713, 118378, 121745, 133274, 164305, 235298, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 524288

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
10069120217 is in the sequence as 10069120217 = 29^6 + 46^6.
139314070233 is in the sequence as 139314070233 = 3^6 + 72^6.
404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.

Cf. A088677 (2 distinct 6th). Supersequence of A106318.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

With[{k = 6}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[8], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999  *)

Removed incorrect program. David A. Corneth, Aug 01 2020

A024670 Numbers that are sums of 2 distinct positive cubes.

Original entry on oeis.org

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674

Offset: 1

Clark Kimberling

nonn

This sequence contains no primes since x^3+y^3=(x^2-x*y+y^2)*(x+y). - M. F. Hasler, Apr 12 2008
There are no terms == 3, 4, 5 or 6 mod 9. - Robert Israel, Oct 07 2014
a(n) mod 2: {1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,1,0, ...} - Daniel Forgues, Sep 27 2018

			9 is in the sequence since 2^3 + 1^3 = 9.
35 is in the sequence since 3^3 + 2^3 = 35.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..902 from M. F. Hasler)
Index to sequences related to sums of cubes

See also: Sums of 2 positive cubes (not necessarily distinct): A003325. Sums of 3 distinct positive cubes: A024975. Sums of distinct positive cubes: A003997. Sums of 2 distinct nonnegative cubes: A114090. Sums of 2 nonnegative cubes: A004999. Sums of 2 distinct positive squares: A004431. Cubes: A000578.
Cf. A373971 (characteristic function).
Indices of nonzero terms in A025468.

N:= 10000: # to get all terms <= N
S:= select(`<=`,{seq(seq(i^3 + j^3, j = 1 .. i-1), i = 2 .. floor(N^(1/3)))},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list));
# Robert Israel, Oct 07 2014

lst={};Do[Do[x=a^3;Do[y=b^3;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/3)],a+1,-1}],{a,Floor[n^(1/3)],1,-1}],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Select[Range@ 1700, Total@ Boole@ Map[And[! MemberQ[#, 0], UnsameQ @@ #] &, PowersRepresentations[#, 2, 3]] > 0 &] (* Michael De Vlieger, May 13 2017 *)

isA024670(n)=for( i=ceil(sqrtn( n\2+1,3)),sqrtn(n-.5,3), isA000578(n-i^3) & return(1)) /* One could also use "for( i=2,sqrtn( n\2-1,3),...)" but this is much slower since there are less cubes in [n/2,n] than in [1,n/2]. Replacing the -1 here by +.5 would yield A003325, allowing for a(n)=x^3+x^3. Replacing -1 by 0 may miss some a(n) of this form due to rounding errors. - M. F. Hasler, Apr 12 2008 */

from itertools import count, takewhile
def aupto(limit):
    cbs = list(takewhile(lambda x: x <= limit, (i**3 for i in count(1))))
    sms = set(c+d for i, c in enumerate(cbs) for d in cbs[i+1:])
    return sorted(s for s in sms if s <= limit)
print(aupto(1674)) # Michael S. Branicky, Sep 28 2021

Name edited by Zak Seidov, May 31 2011

A159843 Sums of two rational cubes.

Original entry on oeis.org

1, 2, 6, 7, 8, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 27, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117

Offset: 1

Steven Finch, Apr 23 2009

Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.
The prime elements are listed in A166246. - Max Alekseyev, Oct 10 2009
Alpöge et al. prove 'that the density of integers expressible as the sum of two rational cubes is strictly positive and strictly less than 1.' The authors remark that it is natural to conjecture that these integers 'have natural density exactly 1/2.' - Peter Luschny, Nov 30 2022
Jha, Majumdar, & Sury prove that every nonzero residue class mod p (for prime p) has infinitely many elements, as do 1 and 8 mod 9. - Charles R Greathouse IV, Jan 24 2023
Alpöge, Bhargava, & Shnidman prove that the lower density of this sequence is at least 2/21 and its upper density is at most 5/6. - Charles R Greathouse IV, Feb 15 2023

H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Levent Alpöge, Manjul Bhargava, and Ari Shnidman, Integers expressible as the sum of two rational cubes, arXiv:2210.10730 [math.NT], Oct. 2022.
Bogdan Grechuk, Existence of Non-Trivial Solutions to Homogeneous Equations, Polynomial Diophantine Equations, Springer, Cham (2024), Chapter 6, 473-536.
Somnath Jha, Dipramit Majumdar, and B. Sury, Infinitely many primes in each of the residue classes 1 and 8 modulo 9 are sums of two rational cubes, arXiv preprint arXiv:2301.06970 [math.NT], 2023-2024.
Index entries for sequences related to sums of cubes

Complement of A185345.
Subsequences include A045980, A004999, and A003325.
Cf. A020894, A020895, A020897, A020898.

(* A naive program with a few pre-computed terms *) nmax = 117; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Join[{1}, Reap[ Do[n = CubeFreePart[x*y*(x + y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union]; A159843 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* Jean-François Alcover, Apr 03 2012 *)

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<3 || mwr[1], return(1)); if(mwr[2]<1, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(ar)); for(effort=1,99, mwr=ellrank(eri,effort); if(mwr[1]>0, return(1), mwr[2]<1, return(0))); "yes under BSD conjecture" \\ Charles R Greathouse IV, Dec 02 2022

A cubefree integer c>2 is in this sequence iff the elliptic curve y^2=x^3+16*c^2 has positive rank. - Max Alekseyev, Oct 10 2009

A004825 Numbers that are the sum of at most 3 positive cubes.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 16, 17, 24, 27, 28, 29, 35, 36, 43, 54, 55, 62, 64, 65, 66, 72, 73, 80, 81, 91, 92, 99, 118, 125, 126, 127, 128, 129, 133, 134, 136, 141, 152, 153, 155, 160, 179, 189, 190, 192, 197, 216, 217, 218, 224, 225, 232, 243, 244, 250, 251, 253

Offset: 1

N. J. A. Sloane

nonn

Or: numbers which are the sum of 3 (not necessarily distinct) nonnegative cubes. - R. J. Mathar, Sep 09 2015

Deshouillers, Hennecart, & Landreau conjecture that this sequence has density 0.0999425... = lim_K Sum_{k=1..K} exp(c*rho(k,K)/K^2)/K where c = -gamma(4/3)^3/6 = -0.1186788..., K takes increasing values in A003418 (or, equivalently, A051451), and rho(k0,K) is the number of triples 1 <= k1,k2,k3 <= K such that k0 = k1^3 + k2^3 + k3^3 mod K. - Charles R Greathouse IV, Sep 16 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, On the density of sums of three cubes, ANTS-VII (2006), pp. 141-155.
Index entries for sequences related to sums of cubes

A003072 is a subsequence.
Cf. A004999.
Column k=3 of A336820.

isA004825 := proc(n)
    local x,y,zc ;
    for x from 0 do
        if 3*x^3 > n then
            return false;
        end if;
        for y from x do
            if x^3+2*y^3 > n then
                break;
            else
                zc := n-x^3-y^3 ;
                if zc >= y^3 and isA000578(zc) then
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A004825 := proc(n)
    option remember;
    local a;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA004825(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A004825(n),n=1..100) ; # R. J. Mathar, Sep 09 2015
# second Maple program:
b:= proc(n, i, t) option remember; n=0 or i>0 and t>0
      and (b(n, i-1, t) or i^3<=n and b(n-i^3, i, t-1))
    end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, -1, a(n-1))
      while not b(k, iroot(k, 3), 3) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 16 2016

q=7; imax=q^3; Select[Union[Flatten[Table[x^3+y^3+z^3, {x,0,q}, {y,x,q}, {z,y,q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

list(lim)=my(v=List(),k,t); for(x=0,sqrtnint(lim\=1,3), for(y=0, min(sqrtnint(lim-x^3,3),x), k=x^3+y^3; for(z=0,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

A045980 Numbers of the form x^3 + y^3 or x^3 - y^3.

Original entry on oeis.org

0, 1, 2, 7, 8, 9, 16, 19, 26, 27, 28, 35, 37, 54, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 128, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 432

Offset: 1

N. J. A. Sloane

Sums of two integer cubes. - Charles R Greathouse IV, Mar 30 2022

			7 = (2)^3 + (-1)^3.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 86.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
M. Kim, Diophantine equations in two variables, arXiv:math/0210329 [math.NT], 2002.
Index entries for sequences related to sums of cubes

A004999 and A003325 are subsequences.

Cf. A222304, A222305, A222306, A027750, A010052.

a045980 n = a045980_list !! (n-1)
a045980_list = 0 : filter f [1..] where
   f x = g $ takeWhile ((<= 4 * x) . (^ 3)) $ a027750_row x where
     g [] = False
     g (d:ds) = r == 0 && a010052 (d ^ 2 - 4 * y) == 1 || g ds
       where (y, r) = divMod (d ^ 2 - div x d) 3
-- Reinhard Zumkeller, Dec 20 2013

Union[Select[Sort[Flatten[Table[{j^3-i^3, j^3+i^3}, {i, 0, 20}, {j, i, 20}]]], #<20^3-19^3&]]
With[{nn=20},Take[Union[Select[Flatten[{Total[#],#[[1]]-#[[2]]}&/@(Tuples[ Range[0,nn],2]^3)],#>-1&]],3*nn]] (* Harvey P. Dale, Jun 22 2014 *)

is(n)=fordiv(n,d, my(L=(d^2-n/d)/3); if(denominator(L)==1 && issquare(d^2-4*L), return(1))); 0 \\ Charles R Greathouse IV, Jun 12 2012

list(lim)={
    my(v=List(),x3,t);
    for(x=0,sqrtnint(lim\=1,3),
        x3=x^3;
        for(y=0,min(sqrtnint(lim-x3,3),x),
            listput(v,x3+y^3)
        )
    );
    for(x=2,t=sqrtint(lim\3),
        x3=x^3;
        for(y=sqrtnint(max(0,x3-lim-1),3)+1,x-1,
            listput(v,x3-y^3)
        )
    );
    t=(t+1)^3-t^3;
    if(t<=lim,listput(v,t));
    Set(v);
} \\ Charles R Greathouse IV, Jun 12 2012, updated Jan 13 2022

is(n)=#thue(thueinit(z^3+1),n) \\ Ralf Stephan, Oct 18 2013

A025446 Number of partitions of n into 2 nonnegative cubes.

Original entry on oeis.org

1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 1

Offset: 0

David W. Wilson

nonn

a(1729) = 2, the first point where a value larger than 1 appears, and where this sequence differs from A373972. - Antti Karttunen, Jun 24 2024

			From _Antti Karttunen_, Jun 24 2024: (Start)
8 = 0^3 + 2^3, and as there are no other partitions of 8 into 2 nonnegative cubes, a(8) = 1.
16 = 2^3 + 2^3, and as there are no other partitions of 16 into 2 nonnegative cubes, a(16) = 1.
1729 = 1^3 + 12^3 = 9^3 + 10^3, and as there are no other partitions of 1729 into 2 nonnegative cubes, a(1729) = 2.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..100080
Wikipedia, 1729 (number)

Cf. A010057, A025455, A004999 (indices of nonzero terms), A373972 (their characteristic function).

A025446(n) = if(n<=2, 1, my(s=0, x=sqrtnint(n,3)); forstep(i=x, 0, -1, my(x3=i^3, y3=n-x3); if(y3>x3, return(s), s += ispower(y3, 3)))); \\ Antti Karttunen, Jun 24 2024

a(n) = A010057(n) + A025455(n) = A010057(n) XOR A025455(n). [The latter by Fermat's Last Theorem] - Antti Karttunen, Jun 24 2024

Data section extended up to a(126) and the secondary offset added by Antti Karttunen, Jun 24 2024

A343326 Number of ways to write n as the integral part of (a^3+b^3)/2 + (c^3+d^3)/6, where a,b,c,d are nonnegative integers with a >= max{b,1} and c >= max{d,1}.

Original entry on oeis.org

2, 3, 3, 2, 4, 7, 4, 1, 4, 6, 3, 4, 3, 6, 5, 6, 5, 3, 7, 5, 2, 4, 6, 4, 5, 7, 5, 2, 6, 7, 1, 2, 8, 4, 6, 5, 9, 10, 7, 4, 6, 7, 6, 2, 5, 8, 4, 6, 5, 5, 6, 4, 2, 7, 7, 2, 3, 9, 5, 3, 4, 6, 5, 7, 9, 7, 8, 8, 12, 5, 5, 6, 9, 10, 7, 5, 7, 7, 5, 4, 3, 6, 4, 5, 6, 8, 9, 7, 5, 10, 5, 5, 3, 7, 10, 3, 3, 8, 5, 10, 9

Offset: 0

Zhi-Wei Sun, Apr 11 2021

nonn

Conjecture: a(n) > 0 for any nonnegative integer n.

This has been verified for all n = 0..10^5.

			a(0) = 2 with 0 = floor((1^3+0^3)/2 + (1^3+0^3)/6) = floor((1^3+0^3)/2 + (1^3+1^3)/6).
a(7) = 1 with 7 = floor((3^3+1^3)/2 + (2^3+2^3)/6).
a(30) = 1 with 30 = floor((2^3+2^3)/2 + (5^3+2^3)/6).
a(111) = 1 with 111 = floor((6^3+1^3)/2 + (2^3+2^3)/6).
a(163) = 1 with 163 = floor((6^3+3^3)/2 + (5^3+5^3)/6).
a(219) = 1 with 219 = floor((4^3+0^3)/2 + (10^3+5^3)/6).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

Cf. A000578, A004999, A343368.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
tab={};Do[r=0;Do[If[CQ[6n+s-3(x^3+y^3)-z^3],r=r+1],{s,Boole[n==0],5},{x,1,((6n+s-1)/3)^(1/3)},{y,0,Min[x,((6n+s-1)/3-x^3)^(1/3)]},{z,0,((6n+s-3(x^3+y^3))/2)^(1/3)}];tab=Append[tab,r],{n,0,100}];Print[tab]

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