A025419 -id:A025419 - cp's OEIS frontend

A025469 Number of partitions of n into 3 distinct positive cubes.

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

Offset: 0

David W. Wilson

nonn

In other words, number of solutions to the equation n = x^3 + y^3 + z^3 with x > y > z > 0. - Antti Karttunen, Aug 29 2017

			From _Antti Karttunen_, Aug 29 2017: (Start)
For n = 36 there is one solution: 36 = 27 + 8 + 1, thus a(36) = 1.
For n = 1009 there are two solutions: 1009 = 10^3 + 2^3 + 1^3 = 9^3 + 6^3 + 4^3, thus a(1009) = 2. This is also the first point where sequence attains value greater than one.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..17073
Index entries for sequences related to sums of cubes

Cf. A025465 (not necessarily distinct), A025468, A025419 (greedy inverse).

Cf. A024975 (positions of nonzero terms), A024974 (positions of terms > 1), A025399-A025402.

A025469 := 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+1 do
            if x^3+2*y^3 > n then
                break;
            end if;
            zcu := n-x^3-y^3 ;
            if zcu > y^3 and isA000578(zcu) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc:
seq(A025469(n),n=1..80) ; # R. J. Mathar, Jun 15 2018

Table[Count[IntegerPartitions[n, {3}], ?(And[UnsameQ @@ #, AllTrue[#, IntegerQ[#^(1/3)] &]] &)], {n, 105}] (* _Michael De Vlieger, Aug 29 2017 *)

A025469(n) = { my(s=0); for(x=1,n,if(ispower(x,3),for(y=x+1,n-x,if(ispower(y,3),for(z=y+1,n-(x+y),if((ispower(z,3)&&(x+y+z)==n),s++)))))); (s); }; \\ Antti Karttunen, Aug 29 2017

a(n) = A025465(n) - A025468(n). - Antti Karttunen, Aug 29 2017

A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

6, 62, 5104, 5978882

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025415, A025419, A123075, A146756, A374226, A374228.

A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

6, 188, 768, 1728, 2640, 21120, 42336, 13824, 71280, 5832, 80352, 74088, 425088, 421875, 1058400, 110592, 287496, 46656

Offset: 1

Zhining Yang, Jun 26 2025

a(19) > 2000000, a(20) = 216000, a(22) = 884736.

			a(3)=768, because 768^2 = 54^3 + 59^3 + 61^3 = 40^3 + 62^3 + 66^3 = 24^3 + 40^3 + 80^3 and no integer less than 768 has 3 solutions.

Cf. A024975, A025419, A377444.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 3, 3],
     0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 2000}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 4}][[All, 1]]

a(18) from Chai Wah Wu, Jul 05 2025

A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

3, 6, 16, 12, 27, 63, 38, 24, 94, 18, 123, 42, 93, 75, 141, 48, 66, 36, 153, 60, 140, 96, 279, 114, 200, 138, 410, 174, 72, 126, 186, 168, 204, 150, 108, 426, 132, 220, 418, 246, 498, 736, 144, 120, 294, 306, 210, 666, 282, 378, 252, 770, 216, 460, 462, 534, 180

Offset: 1

Zhining Yang, Jul 03 2025

nonn

			a(3)=16, because 16^6 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3  and no integer less than 16 has 3 solutions.

Chai Wah Wu, Table of n, a(n) for n = 1..104

Cf. A024975, A025419, A377444, A385354, A385565.

s = Table[{k, Length@Select[PowersRepresentations[k^6, 3, 3], 0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 30}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(41)-a(57) from Chai Wah Wu, Jul 07 2025

A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 1729, 5104, 4445, 4509, 4662, 5454, 6210, 9045, 11124, 14967, 17964, 22051, 26209, 32697, 39564, 46908, 56070, 66222, 78912, 92961, 105841, 123732, 143200, 162801, 188154, 212220, 241614, 271405, 307448, 344016, 383607, 428624, 475273, 529830, 586664, 645120

Offset: 1

Ilya Gutkovskiy, Dec 22 2021

nonn

			For n = 2: 1729 = 1^3 + 12^3 = 9^3 + 10^3.
For n = 3: 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Index entries for sequences related to sums of cubes

Cf. A000578, A025401, A025419, A025421, A275154, A350241.

a(16)-a(27) from Michael S. Branicky, Dec 22 2021

More terms from Jinyuan Wang, Dec 30 2021

A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

11, 21, 64, 144, 330, 846, 342, 252, 1331, 1008, 720, 1890, 3780, 729, 4200, 2016, 1000, 216, 6300, 8352, 10800, 12312, 8568, 19440, 8280, 9576, 21204

Offset: 1

Zhining Yang, Jul 03 2025

a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.

			a(3)=64, because 64^4 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3 and no integer less than 64 has 3 solutions.

Cf. A024975, A025419, A377444, A385354, A385566.

s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
      0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(13), a(15), a(19)-a(21) from Chai Wah Wu, Jul 08 2025

a(22)-a(27) from Chai Wah Wu, Jul 18 2025

A255018 Smallest number that is the sum of 3 nonnegative cubes in exactly n ways.

Original entry on oeis.org

4, 0, 216, 5104, 13896, 161568, 1259712, 2016496, 2562624, 14926248, 58995000, 34012224, 150547032, 471960000, 119095488, 1259712000, 952763904, 5159780352, 3974344704, 2176782336, 10077696000, 2985984000, 36330467328, 30723115968, 23887872000, 17414258688, 72825163776, 75686967000

Offset: 0

Alex Ratushnyak, Feb 25 2015

nonn

			a(0) = 4 because the smallest number that cannot be represented as a sum of 3 nonnegative cubes is 4.
a(1) = 0 is the sum of three 0's.
a(2) = 216 = 3^3 + 4^3 + 5^3 = 6^3 + 0 + 0.
a(3) = 5104 = 1 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A000578, A000446, A011541, A025418, A025419.

TOP = 6000000
a = [0]*TOP
for b in range(TOP):
  b3 = b**3
  if b3*3>=TOP: break
  for c in range(b,TOP):
    c3 = b3 + c**3
    if c3>=TOP: break
    for d in range(c,TOP):
      res = c3 + d**3
      if res>=TOP: break
      a[res] += 1
m = max(a)
r = [-1] * (m+1)
for i in range(TOP):
    if r[a[i]]==-1:  r[a[i]]=i
print(r)

More terms from Rémy Sigrist, Jul 14 2020

A347362 Smallest number which can be decomposed into exactly n sums of three distinct positive cubes, but cannot be decomposed into more than one such sum containing the same cube.

Original entry on oeis.org

36, 1009, 12384, 82278, 746992, 5401404, 15685704, 26936064, 137763072, 251066304, 857520000, 618817536, 3032856000, 2050677000, 6100691904, 36013192704, 16405416000, 96569712000, 48805535232, 131243328000, 611996202000, 201153672000

Offset: 1

Gleb Ivanov, Aug 29 2021

No cube should appear in two or more sums. 5104 = 15^3 + 10^3 + 9^3 = 15^3 + 12^3 + 1^3 = 16^3 + 10^3 + 2^3 is not a(3), because 15^3 appears in more than one sum.

			a(1) = 36 = 1^3 + 2^3 + 3^3.
a(2) = 1009 = 1^3 + 2^3 + 10^3 = 4^3 + 6^3 + 9^3.
a(3) = 12384 = 1^3 + 6^3 + 23^3 = 2^3 + 12^3 + 22^3 = 15^3 + 16^3 + 17^3.

Gleb Ivanov, Rust program for large terms.
Gleb Ivanov, Python program.

Cf. A025333, A025419, A025469, A025398.

Cf. A343968, A343967, A345088, A345087, A345119, A345152.

Monitor[Do[k=1;While[Length@Union@Flatten[p=PowersRepresentations[k,3,3]]!=n*3||Length@p!=n||MemberQ[Flatten@p,0],k++];Print@k,{n,10}],k] (* Giorgos Kalogeropoulos, Sep 03 2021 *)

a(13)-a(15) from Jon E. Schoenfield, Sep 02 2021

a(16)-a(22) from Gleb Ivanov, Sep 12 2021

A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

98, 6578, 811538, 5978882, 1289202642, 292965218, 779888018, 5745705602, 105760443698, 49511121842, 1872511131218, 281539574498, 17673688436978

Offset: 1

Ilya Gutkovskiy, Jul 17 2024

a(16) = 7865870969138. - Michael S. Branicky, Jul 23 2024

			a(2) = 6578 = 1^4 + 2^4 + 9^4 = 3^4 + 7^4 + 8^4.
a(3) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.

Cf. A025415, A025419, A230562, A374696.

a(9)-a(12) from Michael S. Branicky, Jul 22 2024

a(13) from Michael S. Branicky, Jul 23 2024

A385409 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^2, where 0 < x < y < z < w has exactly n integer solutions.

Original entry on oeis.org

10, 42, 39, 153, 126, 276, 273, 312, 315, 476, 588, 336, 546, 777, 1053, 756, 1216, 1386, 1560, 1134, 1323, 1488, 1365, 1368, 1344, 1596, 2366, 2496, 2988, 1680, 2548, 1736, 2184, 3003, 3720, 2520, 3185, 3552, 2268, 3564, 4095, 3213, 4578, 4392, 5208, 4004, 4599, 5733

Offset: 1

Zhining Yang, Jun 27 2025

nonn

Conjecture: a(n) exists for all n.

			a(4)=153, because 153^2 = 5^3 + 15^3 + 21^3 + 22^3 = 2^3 + 7^3 + 15^3 + 27^3 = 6^3 + 8^3 + 9^3 + 28^3 = 1^3 + 5^3 + 11^3 + 28^3 and no integer less than 153 has 4 solutions.

Zhining Yang, Table of n, a(n) for n = 1..1341

Cf. A024975, A025419, A377444, A384430, A385354.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 4, 3],
      0 < #[[1]] < #[[2]] < #[[3]] < #[[4]] &]}, {k, 500}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 10}][[All, 1]]

cp's OEIS Frontend

A025469 Number of partitions of n into 3 distinct positive cubes.

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A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

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A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

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A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

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A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

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A255018 Smallest number that is the sum of 3 nonnegative cubes in exactly n ways.

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A347362 Smallest number which can be decomposed into exactly n sums of three distinct positive cubes, but cannot be decomposed into more than one such sum containing the same cube.

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A374693 a(n) is the smallest number which can be represented as the sum of 3 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

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