A003072 -id:A003072 - cp's OEIS frontend

A063923 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has a nontrivial primitive solution in nonnegative integers.

72, 94, 107, 144, 365, 415, 427, 435, 480, 503, 530, 553, 575, 650, 700, 703, 716, 729, 744, 764, 804, 848, 851, 875, 923, 941, 975, 1004, 1006, 1040, 1044, 1235, 1257, 1313, 1327, 1329, 1369, 1392, 1457, 1469, 1504, 1528, 1537, 1575, 1583, 1588, 1596, 1623, 1653, 1685, 1686

Offset: 1

David W. Wilson, Aug 31 2001

nonn

Primitive means a solution for k has gcd(a,b,c,d,e) = 1. [Corrected by Jianing Song, Jan 24 2020]

Nontrivial means at least two of a,b,c,d,e are nonzero. - Jianing Song, Jan 24 2020

			   72^5 = 19^5 + 43^5 + 46^5 + 47^5 +  67^5;
   94^5 = 21^5 + 23^5 + 37^5 + 79^5 +  84^5;
  107^5 =  7^5 + 43^5 + 57^5 + 80^5 + 100^5.

Jianing Song, Table of n, a(n) for n = 1..400 (All terms from James Waldby link)
L. J. Lander, T. R. Parkin and J. L. Selfridge, A Survey of Equal Sums of Like Powers, Mathematics of Computation, Vol. 21, No. 99 (Jul., 1967), pp. 446-459. See Table III p. 449.
James Waldby, A Table of Fifth Powers equal to a Fifth Power
Index to sequences related to Diophantine equations (5,1,5)

Cf. A063922.
For cubes: A003072, A023041, A261029.
For fourth powers: A003828, A175610, A039664, A003294.

144 and 1006 inserted and name simplified by Jianing Song, Jan 24 2020

More terms from Jinyuan Wang, Jan 24 2020

A336725 A(n,k) is the n-th number that is a sum of k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 10, 8, 4, 5, 19, 17, 10, 5, 6, 36, 34, 24, 13, 6, 7, 69, 67, 49, 29, 17, 7, 8, 134, 132, 98, 64, 36, 18, 8, 9, 263, 261, 195, 129, 84, 43, 20, 9, 10, 520, 518, 388, 258, 160, 99, 55, 25, 10, 11, 1033, 1031, 773, 515, 321, 247, 114, 62, 26, 11, 12, 2058, 2056, 1542, 1028, 642, 384, 278, 129, 66, 29, 12

Offset: 1

Alois P. Heinz, Aug 01 2020

			Square array A(n,k) begins:
   1,  2,  3,   4,   5,   6,    7,    8,    9,   10, ...
   2,  5, 10,  19,  36,  69,  134,  263,  520, 1033, ...
   3,  8, 17,  34,  67, 132,  261,  518, 1031, 2056, ...
   4, 10, 24,  49,  98, 195,  388,  773, 1542, 3079, ...
   5, 13, 29,  64, 129, 258,  515, 1028, 2053, 4102, ...
   6, 17, 36,  84, 160, 321,  642, 1283, 2564, 5125, ...
   7, 18, 43,  99, 247, 384,  769, 1538, 3075, 6148, ...
   8, 20, 55, 114, 278, 734,  896, 1793, 3586, 7171, ...
   9, 25, 62, 129, 309, 797, 2193, 2048, 4097, 8194, ...
  10, 26, 66, 164, 340, 860, 2320, 6568, 4608, 9217, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-11 give: A000027, A000404, A003072, A003338, A003350, A003362, A003374, A003386, A003398, A004810, A004822.
Rows n=1-3 give: A000027, A052944, A145071.
Main diagonal gives A000337.
Cf. A336820.

A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end,
      proc(n, k) option remember; local b; b:=
        proc(x, y) option remember; `if`(x=0, {0}, `if`(y<1, {},
          {b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]}))
        end;
        while nops(w(k)) < n do forget(b);
          l(k):= [l(k)[], (nops(l(k))+1)^k];
          w(k):= sort([select(h-> h

nmax = 12;
pow[n_, k_] := IntegerPartitions[n, {k}, Range[n^(1/k) // Ceiling]^k];
col[k_] := col[k] = Reap[Module[{j = k, n = 1, p}, While[n <= nmax, p = pow[j, k]; If[p =!= {}, Sow[j]; n++]; j++]]][[2, 1]];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2020 *)

A025395 Numbers that are the sum of 3 positive cubes in exactly 1 way.

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, 216, 218, 225, 232, 244, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434, 440

Offset: 1

David W. Wilson

A025456(a(n)) = 1. - Reinhard Zumkeller, Apr 23 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubic Number.
Index entries for sequences related to sums of cubes

Cf. A003072, A024981, A025321, A025396, A025403, A338667, A344188.

Reap[For[n = 1, n <= 500, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 1, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

A057904 Positive integers that are not the sum of exactly three positive cubes.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Eric W. Weisstein

nonn

Differs from A047318 = numbers not congruent to 3 modulo 7: for example, A047318(26) = 29 is not in this sequence. - M. F. Hasler, Jun 30 2025

			3 = 1^3 + 1^3 + 1^3, therefore 3 is not in this sequence. Similarly,
10 = 1^3 + 1^3 + 2^3, therefore 10 is not in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003072 (complement).

Cf. A047318 (not congruent to 3 mod 7), A308065 (not the same).

Select[Range[100], Count[ PowersRepresentations[#, 3, 3], pr_List /; FreeQ[pr, 0]] == 0 &] (* Jean-François Alcover, Oct 31 2012 *)

select( {is_A057904(n)=n<3 || !for(c=sqrtnint(n\/3,3),sqrtnint(n-2,3), isA003325(n-c^3)&&return)}, [1..99]) \\ M. F. Hasler, Jun 30 2025

A025456(a(n)) = 0. - Reinhard Zumkeller, Apr 23 2009

A119977 Triangular numbers that can be written as sum of three positive cubes.

Original entry on oeis.org

3, 10, 36, 55, 66, 136, 153, 190, 253, 378, 496, 528, 820, 946, 1035, 1128, 1485, 3240, 3403, 3655, 4950, 5886, 6903, 7750, 8128, 8256, 9316, 10440, 12403, 13203, 13861, 14365, 14535, 15051, 15753, 16290, 17020, 17205, 17578, 18915

Offset: 1

Reinhard Zumkeller, Jun 03 2006

nonn

Intersection of A003072 and A000217.

			153 = 17*(17+1)/2 = 5^3 + 3^3 + 1^3, therefore 153 is a term.

Donovan Johnson, Table of n, a(n) for n = 1..6577
Index entries for sequences related to sums of cubes

Cf. A113958, A003108, A000578.

Lim=20000;Tlim=Sqrt[2Lim];Clim=Lim^(1/3);Select[Table[n(n+1)/2,{n,Tlim}],MemberQ[Total/@Tuples[Range[Clim]^3,3],#]&] (* James C. McMahon, Sep 23 2024 *)

Two duplicated terms removed by Donovan Johnson, Apr 19 2011

A267414 Integers k such that there exist nonnegative integers x,y,z with k! = x^3 + y^3 + z^3.

Original entry on oeis.org

0, 1, 2, 4, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 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, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Altug Alkan, Jan 14 2016

nonn

From Altug Alkan, David A. Corneth and Chai Wah Wu, Aug 09-26 2020: (Start)
Conjecture I: The natural density of this sequence is 1.
Conjecture II: All integers > 13 are terms. The decomposition is not necessarily unique; for instance, 12! = 35^3 + 309^3 + 766^3 = 240^3 + 504^3 + 696^3.
Deshouillers, Hennecart, & Landreau conjecture (the DHL conjecture) that the sequence of numbers that are a sum of at most three cubes has density 0.0999425... (see links).
This lets us make a heuristic argument that all integers k > 13 are terms.
It was verified for k < 34. For k >= 34 we can use the fact that m is a term if m!/t^3 is the sum of three nonnegative cubes. The cubefree part of 34! is 2686295049620 (cf. A145642) and tau((34!/2686295049620)^(1/3)) = 792 (cf. A248780). 132 terms of corresponding 792 numbers are congruent to 4 or 5 mod 9, that is, there cannot be the sum of three cubes in these 132 terms by modular restriction. So we can see that if 34! isn't the sum of at most three cubes then 792 - 132 = 660 candidate numbers aren't the sum of at most three cubes.
So roughly, if the DHL conjecture holds and if that density can be used as a probability that holds independently for candidates then we have the probability that 34! is the sum of at most 3 cubes to be 1 - (1-0.0999425)^660 ~= 1 - 6.6*10^-31. For larger k this probability doesn't tend to decrease. (End)

			0 and 1 are terms because 0! = 1! = 1 = 0^3 + 0^3 + 1^3.
2 is a term because 2! = 2 = 0^3 + 1^3 + 1^3.
4 is a term because 4! = 24 = 2^3 + 2^3 + 2^3.
From _Chai Wah Wu_, Jan 18 2016: (Start)
9! = 36^3 + 52^3 + 56^3
10! = 4^3 + 96^3 + 140^3
11! = 105^3 + 222^3 + 303^3
12! = 35^3 + 309^3 + 766^3
14! = 135^3 + 3153^3 + 3822^3
15! = 1092^3 + 2040^3 + 10908^3
16! = 7644^3 + 21192^3 + 22212^3
17! = 9984^3 + 22848^3 + 69984^3
18! = 18900^3 + 54060^3 + 184080^3
19! = 131040^3 + 331200^3 + 436320^3
20! = 87490^3 + 1034430^3 + 1098440^3
21! = 59850^3 + 2072070^3 + 3481380^3 (End)
22! = 286272^3 + 8168832^3 + 8334144^3. - _Altug Alkan_, Aug 08 2020
From _Chai Wah Wu_, Aug 09 2020: (Start)
23! = 8255520^3 + 10856160^3 + 28848960^3
24! = 8648640^3 + 9918720^3 + 85216320^3
25! = 31449600^3 + 194947200^3 + 200592000^3
26! = 133526400^3 + 232377600^3 + 729590400^3
27! = 400579200^3 + 697132800^3 + 2188771200^3
28! = 745516800^3 + 3859430400^3 + 6274195200^3
29! = 6029402400^3 + 7705152000^3 + 20136664800^3
30! = 24051081600^3 + 35394105600^3 + 59154883200^3
31! = 63842385600^3 + 74054736000^3 + 196233710400^3
32! = 19948723200^3 + 392984524800^3 + 587164032000^3
33! = 757780531200^3 + 1319649408000^3 + 1812063052800^3
34! = 2423348928000^3 + 5068495555200^3 + 5322645820800^3
35! = 221937408000^3 + 1100266675200^3 + 21780043084800^3
36! = 37944351244800^3 + 43054819315200^3 + 61932511872000^3 (End)
From _Altug Alkan_, Aug 15-26 2020: (Start)
37! = 24795996825600^3 + 74281492454400^3 + 237157683840000^3.
38! = 117664241587200^3 + 120627079372800^3 + 803958680448000^3.
39! = 863357752857600^3 + 953842592102400^3 + 2663078850432000^3.
40! = 2918729189376000^3 + 5087164642560000^3 + 8703942863616000^3.
41! = 7755318514944000^3 + 8120284204032000^3 + 31896357292800000^3.
42! = 89122911958080000^3 + 33781805785728000^3 + 87002517970368000^3.
43! = 122523857584128000^3 + 202407941159424000^3 + 369098064631296000^3.
44! = 259725052274688000^3 + 793899570207744000^3 + 1288734012453888000^3.
45! = 406827658382745600^3 + 1201813420282675200^3 + 4902359567603097600^3.
47! = 12321320074256793600^3 + 20307078211733913600^3 + 62859559551447859200^3.
48! = 25537325843751321600^3 + 149166695523144499200^3 + 208609080169435545600^3.
50! = 1299690649834536960000^3 + 1575788569801205760000^3 + 2896698799298304000000^3.
52! = 4714930301540659200000^3 + 30326925607072174080000^3 + 37482600824578990080000^3.
57! = 2143437030275189096448000^3 + 18952651629200785047552000^3 + 32303499916146500321280000^3. (End)
From _Altug Alkan_, Mar 05-13 2021: (Start)
46! = 5577191426219212800^3 + 6443840881904025600^3 + 17169667908109516800^3.
49! = 671664000771219456000^3 + 662061074870587392000^3 + 247029110344912896000^3.
51! = 9256160466097459200000^3 + 9117812465538416640000^3 + 428071307793592320000^3.
53! = 162171341319623860224000^3 + 14768160510292180992000^3 + 18786201326150049792000^3.
54! = 545218231179130629120000^3 + 335022509605704560640000^3 + 314703105438452290560000^3.
55! = 1946744272579774187520000^3 + 1230901820453108643840000^3 + 1511561473478381445120000^3.
58! = 52226010170722243215360000^3 + 102552481007618403041280000^3 + 104144718055889686855680000^3.
59! = 496516081488480416563200000^3 + 247419327579970911805440000^3 + 104213060097975874805760000^3. (49,51,53,54,55,59 found by _Bernard Landreau_, Mar 05-10 2021) (End)
From _Bernard Landreau_, Feb 10 2023: (Start)
56! = 8440722823838300835840000^3 + 1539870961334538792960000^3 + 4732343335270526976000000^3.
60! = 1954690295686184458321920000^3 + 187526160279422365040640000^3 + 945736839075280596664320000^3.
61! = 6987261145735262954225664000^3 + 5500819928796737985183744000^3 + 3511150067368879423488000^3.
62! = 28126020674003772660940800000^3 + 12303713179773215087247360000^3 + 19449735813987841779056640000^3.
63! = 106514918440099777554186240000^3 + 49252742968526796125306880000^3 + 86830960771932156207267840000^3.
64! = 426059673760399110216744960000^3 + 197010971874107184501227520000^3 + 347323843087728624829071360000^3.
65! = 1825857768347463635450265600000^3 + 1233646969650476271309619200000^3 + 656708896142403679243468800000^3.
66! = 7629164545500731715435233280000^3 + 383304147481048793646366720000^3 + 4645292541653757960968601600000^3.
67! = 32138800724565658662277939200000^3 + 3987806882839318432102809600000^3 + 14753675466796017234670387200000^3.
68! = 121268519043338230583014195200000^3 + 74635666310379772757724364800000^3 + 65491151303650959730645401600000^3.
69! = 440198819826578009858742681600000^3 + 217119306274746004582406553600000^3 + 422815083063767403026566348800000^3. (End)
From _Bernard Landreau_, Apr 12 2023: (Start)
70! = 1684880479643468059918290124800000^3 + 1267939232313822071989803417600000^3 + 1727697134569562112035900620800000^3.
71! = 7869526037543841297006565785600000^3 + 4179944826601729536159999590400000^3 + 6619802079654886665835708416000000^3.
72! = 13437726338581697013357713817600000^3 + 10167574949678977741805794099200000^3 + 38654599603517743131172247961600000^3.
73! = 96869296261623898801464382586880000^3 + 80774308520159270283270497894400000^3 + 144769602970826932947390114693120000^3.
74! = 649373800890254088606178494873600000^3 + 363407978539450964422332584755200000^3 + 207722030872866958396078844313600000^3.
75! = 2347486647113944742227212238848000000^3 + 2199783184771995658848232636416000000^3 + 1070862876804260107568106602496000000^3.
76! = 12262054139494209011130556907520000000^3 + 2762109848253646350901295382528000000^3 + 2746796636906395254645335359488000000^3.
77! = 47421174895780818749100971655168000000^3 + 18679208068237422355741320413184000000^3 + 31756770658228697228286202871808000000^3.
78! = 141193533844368458064892797124608000000^3 + 108335094312749634096990256889856000000^3 + 193437233894764827340173357613056000000^3.
79! = 897795952124597047877074078334976000000^3 + 151955762572905091739065815367680000000^3 + 551184446076431732583718393774080000000^3.
80! = 3554290394480645556188266337402880000000^3 + 1989394527958598219192394328571904000000^3 + 2658759141945971588173630544019456000000^3. (End)

Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, On the density of sums of three cubes, ANTS-VII (2006), pp. 141-155.

Cf. A000142, A000578, A003072, A003325, A004825, A226955.

isA267414 := proc(n)
    local nf,x,y ;
    nf := n! ;
    for x from 0 do
        if 3*x^3 > nf then
            return false;
        end if;
        for y from x do
            if x^3+2*y^3 > nf then
                break;
            end if;
            if isA000578(nf-x^3-y^3) then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 0 to 1000 do
    if isA267414(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jan 23 2016

a(51)-a(64) from Bernard Landreau, Feb 10 2023

a(65)-a(75) from Bernard Landreau, Apr 12 2023

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

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)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

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

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).

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

A024981 Numbers that are the sum of 3 positive cubes, including repetitions.

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

Offset: 1

Clark Kimberling

			An example of repetition: 251 shows twice, because 251 = 1^3+5^3+5^3 = 2^3+3^3+6^3. [_Jean-François Alcover_, Jul 31 2013]

H. Davenport, Sums of three positive cubes, J. London Math. Soc., 25 (1950), 339-343. Coll. Works III p. 999.

Bruno Berselli, Table of n, a(n) for n = 1..2000
Index entries for sequences related to sums of cubes

Cf. A003072, A003325.

m = 8; Sort[Select[Flatten[Table[x^3 + y^3 + z^3, {x, 1, m}, {y, x, m}, {z, y, m}]], # <= m^3 + 2 &]] (* T. D. Noe, Jul 30 2013 *)
max = 500; pr = Table[ PowersRepresentations[n, 3, 3], {n, 1, max}] // Flatten[#, 1]& // Select[#, Times @@ # != 0 &]&; Total[#^3] & /@ pr (* Jean-François Alcover, Jul 31 2013 - replaced my previous incorrect code *)

Corrected by David W. Wilson, May 15 1997

Inserted a second 251 from T. D. Noe, Jul 30 2013

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