A003072 -id:A003072 - cp's OEIS frontend

A004803 Numbers that are the sum of 3 nonzero 10th powers.

3, 1026, 2049, 3072, 59051, 60074, 61097, 118099, 119122, 177147, 1048578, 1049601, 1050624, 1107626, 1108649, 1166674, 2097153, 2098176, 2156201, 3145728, 9765627, 9766650, 9767673, 9824675, 9825698, 9883723, 10814202, 10815225, 10873250

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)
17258390288153 is in the sequence as 17258390288153 = 14^10 + 14^10 + 21^10.
42930989049225 is in the sequence as 42930989049225 = 19^10 + 20^10 + 22^10.
323760702520401 is in the sequence as 323760702520401 = 23^10 + 26^10 + 26^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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).

kmax = 9*10^15; (* max term *)
m = kmax^(1/10) // Ceiling;
Table[k = x^10 + y^10 + z^10; If[k <= kmax, k, Nothing], {x, 1, m}, {y, x, m}, {z, y, m}] // Flatten // Union (* Jean-François Alcover, Jul 19 2017, updated May 02 2023 *)

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

A004804 Numbers that are the sum of 4 nonzero 10th powers.

Original entry on oeis.org

4, 1027, 2050, 3073, 4096, 59052, 60075, 61098, 62121, 118100, 119123, 120146, 177148, 178171, 236196, 1048579, 1049602, 1050625, 1051648, 1107627, 1108650, 1109673, 1166675, 1167698, 1225723, 2097154, 2098177, 2099200, 2156202, 2157225

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)
65969099123 is in the sequence as 65969099123 = 7^10 + 7^10 + 9^10 + 12^10.
1099804917226 is in the sequence as 1099804917226 = 4^10 + 5^10 + 7^10 + 16^10.
1164925542026 is in the sequence as 1164925542026 = 5^10 + 9^10 + 12^10 + 16^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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).

k = 4; p = 10; amax = 3*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j-1], bmax}, {j, 1, k}]]] // Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

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

A004805 Numbers that are the sum of 5 positive 10th powers.

Original entry on oeis.org

5, 1028, 2051, 3074, 4097, 5120, 59053, 60076, 61099, 62122, 63145, 118101, 119124, 120147, 121170, 177149, 178172, 179195, 236197, 237220, 295245, 1048580, 1049603, 1050626, 1051649, 1052672, 1107628, 1108651, 1109674, 1110697, 1166676, 1167699, 1168722, 1225724, 1226747

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)
10352707051 is in the sequence as 10352707051 = 1^10 + 5^10 + 6^10 + 7^10 + 10^10.
59130893253 is in the sequence as 59130893253 = 7^10 + 9^10 + 9^10 + 11^10 + 11^10.
69011865378 is in the sequence as 69011865378 = 6^10 + 6^10 + 9^10 + 9^10 + 12^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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).

k = 5; p = 10; amax = 2*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j - 1], bmax}, {j, 1, k}]]] //Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

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

A004814 Numbers that are the sum of 3 positive 11th powers.

Original entry on oeis.org

3, 2050, 4097, 6144, 177149, 179196, 181243, 354295, 356342, 531441, 4194306, 4196353, 4198400, 4371452, 4373499, 4548598, 8388609, 8390656, 8565755, 12582912, 48828127, 48830174, 48832221, 49005273, 49007320, 49182419, 53022430

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)
204800049005272 is in the sequence as 204800049005272 = 3^11 + 5^11 + 20^11.
2518268235958260 is in the sequence as 2518268235958260 = 16^11 + 19^11 + 25^11.
3786934745885995 is in the sequence as 3786934745885995 = 10^11 + 19^11 + 26^11. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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).

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

A024975 Sums of three distinct positive cubes.

Original entry on oeis.org

36, 73, 92, 99, 134, 153, 160, 190, 197, 216, 225, 244, 251, 281, 288, 307, 342, 349, 352, 368, 371, 378, 405, 408, 415, 434, 469, 476, 495, 521, 532, 540, 547, 560, 567, 577, 584, 586, 603, 623, 638, 645, 664, 684, 701, 729, 736, 738, 755, 757, 764, 792, 794, 801, 820

Offset: 1

Clark Kimberling

nonn

Subsequence of A003072. Equals A024973 if duplicates of repeated entries are removed. - R. J. Mathar, Apr 13 2008

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A122723 (primes in here), A025399-A025402, A025411 (4 distinct positive cubes).

Total/@Subsets[Range[10]^3,{3}]//Union (* Harvey P. Dale, Aug 22 2021 *)

list(lim)=my(v=List(),x3,t); lim\=1; for(x=3,sqrtnint(lim-9,3), x3=x^3; for(y=2,min(x-1,sqrtnint(lim-x3-1,3)), t=x3+y^3; for(z=1,min(y-1,sqrtnint(lim-t,3)), listput(v,t+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 20 2016

{n: A025469(n) >= 1}. - R. J. Mathar, Jun 15 2018

Verified by Don Reble, Nov 19 2006

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

A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

Original entry on oeis.org

6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108, 110, 111, 112, 113

Offset: 1

N. J. A. Sloane

nonn

Numbers w such that w^3 = x^3+y^3+z^3, x>y>z>=0, is soluble.
A226903(n) + 1 is an infinite subsequence parametrized by Shiraishi in 1826. - Jonathan Sondow, Jun 22 2013
Because of Fermat's Last Theorem, sequence lists numbers w such that w^3 = x^3+y^3+z^3, x>y>z>0, is soluble. In other words, z cannot be 0 because x and y are positive integers by definition of this sequence. - Altug Alkan, May 08 2016
This sequence is the same as numbers w such that w^3 = x^3+y^3+z^3, x>=y>=z>0, is soluble as Legendre showed that a^3+b^3=2*c^3 only has the trivial solutions a = b or a = -b (see Dickson's History of the Theory of Numbers, vol. II, p. 573). - Chai Wah Wu, May 13 2017

			20 belongs to the sequence as 20^3 = 7^3 + 14^3 + 17^3.

Ya. I. Perelman, Algebra can be fun, pp. 142-143.

Nathaniel Johnston and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (1..1500 from Nathaniel Johnston)
A. Russell and C. E. Gwyther, The Partition of Cubes, The Mathematical Gazette, 21 (1937), pp. 33-35.
Index to sequences related to Diophantine equations (3,1,3)

Cf. A001235, A003072, A114923, A225908, A226903.

for w from 1 to 113 do for z from 0 to w-1 do bk:=0: for y from z+1 to w-1 do for x from y+((w+z) mod 2) to w-1 by 2 do if(x^3+y^3+z^3=w^3)then printf("%d, ",w); bk:=1: break: fi: od: if(bk=1)then break: fi: od: if(bk=1)then break: fi: od: od: # Nathaniel Johnston, Jun 22 2013

Select[Range[200], n |-> Length[PowersRepresentations[n^3, 3, 3]] > 1] (* Paul C Abbott, May 07 2025 *)

has(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
list(lim)=my(v=List(),a3,t); lim\=1; for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim, sqrtnint(a3-lim-1,3)+1,1), a-1, t=a3-b^3; if(has(t), listput(v,a)))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

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

A008917 Numbers that are the sum of 3 positive cubes in more than one way.

Original entry on oeis.org

251, 1009, 1366, 1457, 1459, 1520, 1730, 1737, 1756, 1763, 1793, 1854, 1945, 2008, 2072, 2241, 2414, 2456, 2458, 2729, 2736, 3060, 3391, 3457, 3592, 3599, 3655, 3745, 3926, 4105, 4112, 4131, 4168, 4229, 4320, 4376, 4402, 4437, 4447

Offset: 1

N. J. A. Sloane and J. H. Conway

Of course reordering the terms does not count.

A025456(a(n)) > 1. [Reinhard Zumkeller, Apr 23 2009]

			a(2) = 1009 = 1^3+2^3+10^3 = 4^3+6^3+9^3.

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Christian N. K. Anderson, Decomposition of first 10000 terms into multiple cube triples.

Cf. A001235, A003072, A024796, A025396, A025398, A025406, A309762.

Select[Range[4450], 1 < Length @ Cases[PowersRepresentations[#, 3, 3], {?Positive, ?Positive, ?Positive}] &]  (* _Jean-François Alcover, Apr 04 2011 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1, k, for(b=a, k, for(c=b, k, if(a^3+b^3+c^3==n, d++)))); d
n=3; while(n<5000, if(is(n)>1, print1(n, ", ")); n++) \\ Derek Orr, Aug 27 2015

A007490 Primes of form x^3 + y^3 + z^3 where x,y,z > 0.

Original entry on oeis.org

3, 17, 29, 43, 73, 127, 179, 197, 251, 277, 281, 307, 349, 359, 397, 433, 521, 547, 557, 577, 593, 701, 757, 811, 853, 857, 863, 881, 919, 953, 1009, 1051, 1091, 1217, 1249, 1367, 1459, 1483, 1559, 1583, 1637, 1753, 1801, 1861, 1907, 2017, 2027, 2069, 2087

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Heath-Brown shows that this sequence is infinite. - Charles R Greathouse IV, Jul 23 2009

The definition implies x, y, z > 0, so the representation (x=0, y=z=1) for the prime 2 or the representation (x=-4, y=-2, z=5) for the prime 53 are not admitted. - R. J. Mathar, Mar 19 2010

W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 108.
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..10000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
R. G. Wilson, V, Note, n.d.

Cf. A003072 (all numbers).

nn = 3000; Select[Union[Flatten[Table[x^3 + y^3 + z^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}, {z, y, (nn - x^3 - y^3)^(1/3)}]]], PrimeQ] (* T. D. Noe, Sep 18 2012 *)

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), if(isprime(t=k+z^3), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

More terms from Vladimir Joseph Stephan Orlovsky, Mar 18 2010

Definition clarified by Charles R Greathouse IV, Sep 14 2015

cp's OEIS Frontend

A004803 Numbers that are the sum of 3 nonzero 10th powers.

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A004804 Numbers that are the sum of 4 nonzero 10th powers.

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A004805 Numbers that are the sum of 5 positive 10th powers.

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A004814 Numbers that are the sum of 3 positive 11th powers.

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A024975 Sums of three distinct positive cubes.

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

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A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

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A025456 Number of partitions of n into 3 positive cubes.