A057904 -id:A057904 - cp's OEIS frontend

A003072 Numbers that are the sum of 3 positive cubes.

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, 216, 218, 225, 232, 244, 251, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434

Offset: 1

N. J. A. Sloane, David W. Wilson

A119977 is a subsequence; if m is a term then there exists at least one k>0 such that m-k^3 is a term of A003325. - Reinhard Zumkeller, Jun 03 2006
A025456(a(n)) > 0. - Reinhard Zumkeller, Apr 23 2009
Davenport proved that a(n) << n^(54/47 + e) for every e > 0. - Charles R Greathouse IV, Mar 26 2012

			a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.

K. D. Bajpai, Table of n, a(n) for n = 1..12955 (first 1000 terms from T. D. Noe)
H. Davenport, Sums of three positive cubes, J. London Math. Soc., 25 (1950), 339-343. Coll. Works III p. 999.
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Subsequence of A004825.
Cf. A003325, A024981, A057904 (complement), A010057, A000578, A023042 (subsequence of cubes).
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
- squares: A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2);
- cubes: A003325 (2, 3), A003072 (3, 3), 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);
- fourth powers: 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);
- fifth powers: 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);
- sixth powers: 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);
- seventh powers: 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);
- eighth powers: A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003386 (8, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8);
- ninth powers: 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);
- tenth powers: 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);
- eleventh powers: 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).

a003072 n = a003072_list !! (n-1)
a003072_list = filter c3 [1..] where
   c3 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (> 0) $ map (x -) $ a003325_list
-- Reinhard Zumkeller, Mar 24 2012

isA003072 := proc(n)
    local x,y,z;
    for x from 1 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;
            end if;
            if isA000578(n-x^3-y^3) then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 1000 do
    if isA003072(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 23 2016

Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jean-François Alcover, Apr 29 2011 *)
With[{upto=500},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2,3]]]^3,3]],#<=upto&]] (* Harvey P. Dale, Oct 25 2021 *)

sum(n=1,11,x^(n^3),O(x^1400))^3 /* Then [i|i<-[1..#%],polcoef(%,i)] gives the list of powers with nonzero coefficient. - M. F. Hasler, Aug 02 2020 */

list(lim)=my(v=List(),k,t); lim\=1; for(x=1,sqrtnint(lim-2,3), for(y=1, min(sqrtnint(lim-x^3-1,3),x), k=x^3+y^3; for(z=1,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

{n: A025456(n) >0}. - R. J. Mathar, Jun 15 2018

Incorrect program removed by David A. Corneth, Aug 01 2020

A025456 Number of partitions of n into 3 positive cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

If A025455(n) > 0 then a(n + k^3) > 0 for k>0; a(A119977(n))>0; a(A003072(n))>0. - Reinhard Zumkeller, Jun 03 2006
a(A057904(n))=0; a(A003072(n))>0; a(A025395(n))=1; a(A008917(n))>1; a(A025396(n))=2. - Reinhard Zumkeller, Apr 23 2009
The first term > 1 is a(251) = 2. - Michel Marcus, Apr 23 2019

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Least inverses are A025418.

Cf. A025455, A003108, A003072 (1 or more ways), A008917 (two or more ways), A025395-A025398.

A025456 := proc(n)
    local a,x,y,zcu ;
    a := 0 ;
    for x from 1 do
        if 3*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+2*y^3 > n then
                break;
            end if;
            zcu := n-x^3-y^3 ;
            if isA000578(zcu) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

a[n_] := Count[ PowersRepresentations[n, 3, 3], pr_List /; FreeQ[pr, 0]]; Table[a[n], {n, 0, 107}] (* Jean-François Alcover, Oct 31 2012 *)

a(n)=sum(a=sqrtnint(n\3,3),sqrtnint(n,3),sum(b=1,a,my(C=n-a^3-b^3,c);ispower(C,3,&c)&&0Charles R Greathouse IV, Jun 26 2013

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Second offset from Michel Marcus, Apr 23 2019

A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Wesley Ivan Hurt, Jul 27 2019

nonn

Differs from A047318 (not congruent to 3 (mod 7)) starting at a(n=22) = 24 = 3*7 + 3 which is in this sequence but not in A047318. - M. F. Hasler, Jun 30 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A033950, A309347.

Different from A047318 (not congruent to 3 modulo 7) and A057904.

notref:= proc(n) option remember; n mod numtheory:-tau(n) <> 0 end proc:
filter:= proc(n)
  andmap(t -> notref(t) or notref(n-t) or notref(n-2*t), [$1 .. (n-1)/2])
end proc:
select(filter, [$0..100]); # Robert Israel, Jul 29 2025

Flatten[Table[If[Sum[(1 - Ceiling[(n - 2 i)/DivisorSigma[0, n - 2 i]] + Floor[(n - 2 i)/DivisorSigma[0, n - 2 i]]) (1 - Ceiling[i/DivisorSigma[0, i]] + Floor[i/DivisorSigma[0, i]]) (1 - Ceiling[(n - i)/DivisorSigma[0, n - i]] + Floor[(n - i)/DivisorSigma[0, n - i]]), {i, Floor[(n - 1)/2]}] == 0, n, {}], {n, 0, 100}]]

A135693 Not the sum of three positive squares or cubes.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 15, 23, 31, 87, 111, 119, 148, 167, 263, 311, 335, 391, 407, 455, 559, 599, 839, 951, 1159, 1231, 1287, 1303, 1391, 1455, 1463, 1607, 1660, 1679, 1751, 1863, 1991, 2351, 2615, 2799, 3247, 3983, 4327, 4367, 5199, 5655, 6047, 6159, 6351

Offset: 1

Zak Seidov and Giovanni Resta, Feb 24 2008

nonn

No other terms < 10^9. Presumably the sequence is finite.

Sum can include a mix of squares and cubes. - James C. McMahon, Apr 19 2025

Chai Wah Wu, Table of n, a(n) for n = 1..82

Cf. A113505, A135367.

Cf. A004214, A057904.

lim=40000; s=Range[Sqrt[lim]]^2; c=Range[Surd[lim, 3]]^3; A135693=Complement[Range[lim], Select[Total/@Tuples[Union[s, c], {3}], #<=lim&]] (* James C. McMahon, Apr 19 2025 *)

Edited by N. J. A. Sloane, Mar 01 2008

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A003072 Numbers that are the sum of 3 positive cubes.

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A025456 Number of partitions of n into 3 positive cubes.

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A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.

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A135693 Not the sum of three positive squares or cubes.

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Mathematica

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