A004825 -id:A004825 - 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

A004999 Sums of two nonnegative cubes.

Original entry on oeis.org

0, 1, 2, 8, 9, 16, 27, 28, 35, 54, 64, 65, 72, 91, 125, 126, 128, 133, 152, 189, 216, 217, 224, 243, 250, 280, 341, 343, 344, 351, 370, 407, 432, 468, 512, 513, 520, 539, 559, 576, 637, 686, 728, 729, 730, 737, 756, 793, 854, 855, 945, 1000, 1001

Offset: 1

N. J. A. Sloane, Steven Finch

T. D. Noe, Table of n, a(n) for n = 1..1000
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
Index entries for sequences related to sums of cubes

Subsequence of A045980; A003325 is a subsequence.
Cf. A000578, A004825, A010057, A373972 (characteristic function).
Indices of nonzero terms in A025446.

a004999 n = a004999_list !! (n-1)
a004999_list = filter c2 [1..] where
   c2 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (>= 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Dec 20 2013

Union[(#[[1]]^3+#[[2]]^3)&/@Tuples[Range[0,20],{2}]] (* Harvey P. Dale, Dec 04 2010 *)

is(n)=my(k1=ceil((n-1/2)^(1/3)), k2=floor((4*n+1/2)^(1/3)), L); fordiv(n,d,if(d>=k1 && d<=k2 && denominator(L=(d^2-n/d)/3)==1 && issquare(d^2-4*L), return(1))); 0
list(lim)=my(v=List());for(x=0,(lim+.5)^(1/3),for(y=0,min(x,(lim-x^3)^(1/3)),listput(v,x^3+y^3))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2012

is(n)=my(L=sqrtnint(n-1,3)+1,U=sqrtnint(4*n,3));fordiv(n,m,if(L<=m&m<=U,my(ell=(m^2-n/m)/3);if(denominator(ell)==1&&issquare(m^2-4*ell),return(1))));0 \\ Charles R Greathouse IV, Apr 16 2013

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

A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 3, 5, 5, 0, 1, 2, 3, 8, 8, 6, 0, 1, 2, 3, 4, 9, 9, 7, 0, 1, 2, 3, 4, 16, 10, 10, 8, 0, 1, 2, 3, 4, 5, 17, 16, 13, 9, 0, 1, 2, 3, 4, 5, 32, 18, 17, 16, 10, 0, 1, 2, 3, 4, 5, 6, 33, 19, 24, 17, 11, 0, 1, 2, 3, 4, 5, 6, 64, 34, 32, 27, 18, 12

Offset: 1

Alois P. Heinz, Aug 04 2020

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,   0,   0,   0,  0,  0, ...
   1,  1,  1,  1,  1,  1,   1,   1,   1,  1,  1, ...
   2,  2,  2,  2,  2,  2,   2,   2,   2,  2,  2, ...
   3,  4,  3,  3,  3,  3,   3,   3,   3,  3,  3, ...
   4,  5,  8,  4,  4,  4,   4,   4,   4,  4,  4, ...
   5,  8,  9, 16,  5,  5,   5,   5,   5,  5,  5, ...
   6,  9, 10, 17, 32,  6,   6,   6,   6,  6,  6, ...
   7, 10, 16, 18, 33, 64,   7,   7,   7,  7,  7, ...
   8, 13, 17, 19, 34, 65, 128,   8,   8,  8,  8, ...
   9, 16, 24, 32, 35, 66, 129, 256,   9,  9,  9, ...
  10, 17, 27, 33, 36, 67, 130, 257, 512, 10, 10, ...

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

Columns k=1-11 give: A001477(n-1), A001481, A004825, A004833, A004845, A004857, A004869, A004881, A004893, A004905, A004917.
A(n+j,n) for j=0-3 give: A001477(n-1), A000027, A000079, A000051.
Cf. A336725.

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 or y<1, {},
          {0, 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

b[n_, k_, i_, t_] := b[n, k, i, t] = n == 0 || i > 0 && t > 0 && (b[n, k, i - 1, t] || i^k <= n && b[n - i^k, k, i, t - 1]);
A[n_, k_] := A[n, k] = Module[{m}, For[m = 1 + If[n == 1, -1, A[n - 1, k]], !b[m, k, m^(1/k) // Floor, k], m++]; m];
Table[A[n, 1+d-n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, Dec 03 2020, using Alois P. Heinz's code for columns *)

A(n,k) = n-1 for n <= k+1.

A022555 Positive integers that are not the sum of two nonnegative cubes.

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

N. J. A. Sloane

nonn

Omits the positive cubes (A000578) since m^3 = 0^3 + m^3, so is different from A057903.

Complement of A004999. Cf. A004825, A022561, A022566, A057903, A000578.

read transforms; cub := series(add(q^(n^3),n=0..100),q,1000001); t1 := series(cub^2,q,2000); t2 := POWERS(t1,q,2000); COMPl(t2);

r[n_] := Reduce[x >= 0 && y >= 0 && n == x^3 + y^3, {x, y}, Integers]; Select[ Range[80], r[#] === False &]  (* Jean-François Alcover, Nov 06 2012 *)
Select[Range@100,PowersRepresentations[#,2,3]=={}&] (* A much faster solution given by Giovanni Resta, Nov 06 2012 *)

Edited by N. J. A. Sloane, Sep 28 2007

A022561 Numbers that are not the sum of 3 nonnegative cubes.

Original entry on oeis.org

4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 25, 26, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 63, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

N. J. A. Sloane

nonn

Complement of A004825.

lst1=Range[1359]; lst3=Take[Union[Flatten[Table[a^3+b^3+c^3,{a,0,16},{b,0,16},{c,0,16}]]],234] Complement[lst1,lst3] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)
Select[Range[90], PowersRepresentations[#, 3, 3] == {} &] (* Bruno Berselli, Nov 09 2012, after Giovanni Resta in A022555 *)

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

A336205 Numbers k that can be expressed as x^3 + y^3 + z^3 with x^2 + y^2 + z^2 <= k where x, y, z are integers.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 43, 45, 46, 48, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 71, 72, 73, 80, 81, 83, 88, 90, 91, 92, 97, 98, 99, 100, 101, 106, 109, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 132

Offset: 1

Altug Alkan, Jul 12 2020

See A336240 for border case x^2 + y^2 + z^2 = x^3 + y^3 + z^3.
What is the natural density of this sequence?
There are infinitely many infinite parametric families of solutions which have negative values in (x,y,z). For example, 8*(3*a-1)^2*m^6 + 12*(3*a-1)*(a-1)*m^4 - 6*(2*a-1)*m^2 + 2*a^3 + 1 are terms for all a >= 0, m >= 0. (x = 1 - (6*a-2)*m^2, y = a - m*(1-(6*a-2)*m^2), z = a + m*(1-(6*a-2)*m^2)). - Altug Alkan, Jul 17 2020
By definition, corresponding (x,y,z) variables are produced by equation x^3 + y^3 + z^3 = x^2 + y^2 + z^2 + t with t >= 0. That is, x^2*(x-1) + y^2*(y-1) + z^2*(z-1) >= 0. Conjecture: Every even integer can be represented as x^2*(x-1) + y^2*(y-1) + z^2*(z-1) where x, y, z are integers. - Altug Alkan, Jul 19 2020

			11 is not a term because there is no (x,y,z) with x^2 + y^2 + z^2 <= 11 when x^3 + y^3 + z^3 = 11.
18 is a term because (-1)^3 + (-2)^3 + 3^3 = 18 and (-1)^2 + (-2)^2 + 3^2 <= 18.
61 is a term because (-4)^3 + 0^3 + 5^3 = 61 and (-4)^2 + 0^2 + 5^2 <= 61.
354 is a term because (-11)^3 + (-8)^3 + 13^3 = (-11)^2 + (-8)^2 + 13^2 = 354.

Robert Israel, Table of n, a(n) for n = 1..8000
Rémy Sigrist, C program for A336205

Cf. A004825 (subsequence), A060464 (supersequence), A336240.

C
```
See Links section.
```

filter:= proc(n) local x,y,z,e1,e2;
  for x from 0 while 3*x^2 <= n do
    for y from 0 while x^2 + 2*y^2 <= n do
      for e1 in [-1,1] do for e2 in [-1,1] do
        z:= surd(n + e1*x^3 + e2*y^3,3);
        if z::integer and x^2 + y^2 + z^2 <= n then return true fi;
  od od od od;
  false
end proc:
select(filter, [$0..200]); # Robert Israel, Jul 12 2020

filter[n_] := Module[{x, y, z, e1, e2},
  For[x = 0, 3*x^2 <= n, x++,
    For[y = 0, x^2 + 2*y^2 <= n, y++,
      For[e1 = -1, e1 <= 1, e1 += 2, For[e2 = -1, e2 <= 1, e2 += 2,
        z = (n + e1*x^3 + e2*y^3)^(1/3);
        If[IntegerQ[z] && x^2 + y^2 + z^2 <= n, Return[True]]
  ]]]]; False];
Select[Range[0, 200], filter] (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)

A351179 Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 5, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 1, 1, 6, 3, 3, 3, 1, 1, 1, 5, 3, 3, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 3, 3, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 3, 1, 2, 3, 1, 1, 1, 3, 3, 7, 3, 2, 1, 1

Offset: 0

Zhi-Wei Sun, Feb 04 2022

nonn

a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).

Aside from a(96) = 7 and a(850) = 8, a(n) <= 6 for n <= 10^6. - Charles R Greathouse IV, Feb 10 2022

			a(5) = 3 with 3^6*5 = 2^6 + 5^3 + 12^3 + 12^3.
a(12) = 5 with 5^6*12 = 3^6 + 19^3 + 34^3 + 52^3.
a(22) = 2 with 2^6*22 = 1^6 + 4^3 + 7^3 + 10^3.
a(31) = 6 with 6^6*31 = 0^6 + 4^3 + 15^3 + 113^3.
a(96) = 7 with 7^6*96 = 0^6 + 2^3 + 38^3 + 224^3.
a(101) = 4 with 4^6*101 = 3^6 + 22^3 + 39^3 + 70^3.
a(850) = 8 with 8^6*850 = 5^6 + 508^3 + 442^3 + 175^3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
H. W. Richmond, On analogues of Waring's problem for rational numbers, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Cf. A000578, A001014, A004825, A350714, A351199.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[m=1;Label[bb];k=m^6;Do[If[CQ[k*n-w^6-x^3-y^3],tab=Append[tab,m];Goto[aa]],{w,0,(k*n)^(1/6)},{x,0,((k*n-w^6)/3)^(1/3)},{y,x,((k*n-w^6-x^3)/2)^(1/3)}];
m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

a(n) <= A351199(n)^2. - Charles R Greathouse IV, Feb 05 2022

A336449 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 24^6.

Original entry on oeis.org

1, 9, 16, 25, 49, 81, 121, 169, 225, 289, -356, 361, 441, 529, 625, 729, 841, -948, 961, 1045, 1089, 1225, 1369, 1521, 1681, -1715, 1849, 1876, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, -3587, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5769, 5929

Offset: 1

XU Pingya, Aug 08 2020

sign

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

Let x = a^(2*k) - (a^k)*t - t^2, y = a^(2*k) + (a^k)*t - t^2, z = t^2; then x^3 + y^3 + 2*z^3 = 2*a^(6*k). When a = 4, k = 1, t = 2*n + 1; (x, y, z) are primitive solutions of equation. Thus, terms of A016754 are terms of the sequence.

			(-15)^3 + (-27)^3 + 2*25^3 = 11^3 + (-29)^3 + 2*25^3 = 8192, 25 is a term.
(-65)^3 + (449)^3 + 2*(-356)^3 = 8192, -356 is a term.

R. K. Guy, Unsolved Problems in Number Theory, D5.

Cf. A000290, A000578, A003215, A004825, A004826, A016754, A050791, A130472, A195006, A336450.

Clear[t]
t = {};
Do[y = (8192 - x^3 - 2z^3)^(1/3) /. (-1)^(1/3) -> -1;
If[Abs@x <= Abs@y && IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -5929, 5929}, {x, -Round[(Abs[8192 - 2z^3]/3)^(1/2)], Round[(Abs[8192 - 2z^3]/3)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 12000}];
Select[v, MemberQ[u, #] &]

A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 25^6.

Original entry on oeis.org

1, -3, 4, 9, 16, 25, 36, 49, -56, 64, 81, 88, -104, 121, 144, -167, 169, 177, 196, -203, -243, -255, 256, 277, 289, 324, 361, -363, 373, -395, -411, 441, 484, 529, 576, 676, 709, -719, 729, 784, 841, 961, 1017, 1024, -1028, 1080, 1089, -1091, 1156, 1296, 1369

Offset: 1

XU Pingya, Aug 08 2020

sign

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

Let x = a^(2*m) - (a^m)*t - t^2, y = a^(2*m) + (a^m)*t - t^2, z = t^2; then x^3 + y^3 + 2*z^3 = 2*a^(6*m). When a = 5, m = 1, t = 5*n + k(k = {1, 2, 3, 4}); (x, y, z) are primitive solutions of equation. Thus, A047201(n)^2 are terms of the sequence.

			(-20)^3 + 34^3 + 2*(-3)^3 = 31250, -3 is a term.
(-11)^3 + 29^3 + 2*16^3 = 15^3 + 27^3 + 2*16^3 = 31250, 16 is a term.

R. K. Guy, Unsolved Problems in Number Theory, D5.

Cf. A000290, A000578, A003215, A004825, A004826, A047201, A050791, A130472, A195006, A336449.

Clear[t]
t = {};
Do[y = (31250 - x^3 - 2z^3)^(1/3) /. (-1)^(1/3) -> -1;
If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -1369, 1369}, {x, -Round[(Abs[31250 - 2z^3]/3)^(1/2)], Round[(Abs[31250 - 2z^3]/3)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 2739}];
Select[v, MemberQ[u, #] &]

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A022555 Positive integers that are not the sum of two nonnegative cubes.

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A022561 Numbers that are not the sum of 3 nonnegative cubes.

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

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A336205 Numbers k that can be expressed as x^3 + y^3 + z^3 with x^2 + y^2 + z^2 <= k where x, y, z are integers.

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C

A336449 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 24^6.

A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 25^6.