A096545 -id:A096545 - cp's OEIS frontend

A095867 Values y associated with A096545(n), sorted on z, then on y and finally on x.

4, 6, 14, 10, 19, 17, 15, 32, 30, 36, 17, 23, 34, 42, 19, 51, 54, 38, 61, 39, 43, 59, 60, 23, 33, 48, 53, 55, 48, 54, 69, 43, 54, 31, 40, 38, 82, 53, 70, 75, 74, 86, 95, 96, 92, 31, 84, 51, 94, 47, 34, 55, 51, 65, 85, 76, 57, 123, 73, 121, 81, 108, 64, 71, 73, 135, 75, 107, 87

Offset: 1

Ray Chandler, Jun 28 2004

nonn

For 0

			a(1)=4 corresponding to the quadruple (3,4,5,6).

David A. Corneth, Table of n, a(n) for n = 1..13798 (terms corresponding to z <= 8000)
Fred Richman, Sums of Three Cubes

Primitive quadruples (x, y, z, w) = (A095868, A095867, A096545, A096546).

s[w_] := Solve[0 < x < y < z && x^3 + y^3 + z^3 == w^3 && GCD[x, y, z, w] == 1, {x, y, z}, Integers];
xyzw = Reap[For[w = 1, w <= 200, w++, sw = s[w]; If[sw != {}, Print[{x, y, z, w} /. sw; Sow[{x, y, z, w} /. sw ]]]]][[2, 1]] // Flatten[#, 1]&;
SortBy[xyzw, {#[[3]]&, #[[2]]&, #[[1]]&}][[All, 2]] (* Jean-François Alcover, Mar 06 2020 *)

A095868 Values x associated with A096545(n), sorted on z, then on y and finally on x.

Original entry on oeis.org

3, 1, 7, 3, 18, 4, 11, 6, 27, 3, 2, 16, 29, 15, 12, 22, 7, 36, 50, 34, 38, 58, 19, 14, 31, 25, 28, 26, 38, 20, 45, 21, 32, 25, 17, 25, 15, 19, 33, 29, 50, 23, 86, 94, 19, 12, 49, 13, 23, 16, 3, 9, 44, 13, 72, 5, 38, 69, 44, 3, 12, 107, 31, 1, 71, 1, 22, 96, 65, 48, 69, 48, 46, 59

Offset: 1

Ray Chandler, Jun 28 2004

nonn

For 0

			a(1)=3 corresponding to the quadruple (3,4,5,6).

David A. Corneth, Table of n, a(n) for n = 1..13798 (terms corresponding to z <= 8000)
Fred Richman, Sums of Three Cubes

Primitive quadruples (x, y, z, w) = (A095868, A095867, A096545, A096546).

s[w_] := Solve[0 < x < y < z && x^3 + y^3 + z^3 == w^3 && GCD[x, y, z, w] == 1, {x, y, z}, Integers];
xyzw = Reap[For[w = 1, w <= 200, w++, sw = s[w]; If[sw != {}, Print[{x, y, z, w} /. sw; Sow[{x, y, z, w} /. sw ]]]]][[2, 1]] // Flatten[#, 1]&;
SortBy[xyzw, {#[[3]]&, #[[2]]&, #[[1]]&}][[All, 1]] (* Jean-François Alcover, Mar 06 2020 *)

A096546 Values w associated with A096545(n), sorted on z, then on y and finally on x.

Original entry on oeis.org

6, 9, 20, 19, 28, 25, 29, 41, 46, 46, 41, 44, 53, 58, 54, 67, 70, 69, 85, 72, 75, 90, 82, 71, 76, 81, 84, 87, 87, 87, 97, 88, 93, 88, 89, 90, 108, 96, 105, 110, 113, 116, 134, 139, 122, 103, 121, 108, 126, 111, 115, 120, 123, 127, 141, 132, 129, 160, 137, 159, 145, 171

Offset: 1

Lekraj Beedassy, Jun 25 2004

nonn

For 0

			Entry 87, for example, is associated with primitive quadruples (x, y, z, w)= (26, 55, 78, 87), (38, 48, 79, 87), (20, 54, 79, 87) satisfying x^3 + y^3 + z^3 = w^3, for 0<x<y<z=A096545(n), with n=28, 29, 30.

David A. Corneth, Table of n, a(n) for n = 1..13798 (terms corresponding to z <= 8000)
Ajai Choudhry, On equal sums of cubes, Rocky Mountains Journal of Mathematics, 28 (4), 1998.
Fred Richman, Sums of Three Cubes

Primitive quadruples (x, y, z, w) = (A095868, A095867, A096545, A096546).

s[w_] := Solve[0 < x < y < z && x^3 + y^3 + z^3 == w^3 && GCD[x, y, z, w] == 1, {x, y, z}, Integers];
xyzw = Reap[For[w = 1, w <= 200, w++, sw = s[w]; If[sw != {}, Print[{x, y, z, w} /. sw; Sow[{x, y, z, w} /. sw ]]]]][[2, 1]] // Flatten[#, 1]&;
SortBy[xyzw, {#[[3]]&, #[[2]]&, #[[1]]&}][[All, 4]] (* Jean-François Alcover, Mar 06 2020 *)

From Thomas Scheuerle, Jan 29 2025: (Start)
Ajai Choudhry gave this beautiful solution for this problem:
  v1,v2,v3 are integers > 0, v2 > v1, v3 > floor((v1^3+v2^3)^(1/3)).
  x*d =  v3*( -v2^3 - v1^3 + v3^3).
  y*d = -v2^4 + 2*v2^3*v1 - 3*v2^2*v1^2 + 2*v2*v1^3 - v1^4 + (v1+v2)*v3^3.
  z*d =  v2^4 - 2*v2^3*v1 + 3*v2^2*v1^2 - 2*v2*v1^3 + v1^4 + (2*v2-v1)*v3^3.
  w*d =  v3*( v2^3 + (v2-v1)^3 + v3^3).
  d = gcd(x*r, y*r, z*r, w*r). (End)

Extended by Ray Chandler, Jun 28 2004

A328149 Numbers whose set of divisors contains a quadruple (x, y, z, w) satisfying x^3 + y^3 + z^3 = w^3.

Original entry on oeis.org

60, 72, 120, 144, 180, 216, 240, 288, 300, 360, 420, 432, 480, 504, 540, 576, 600, 648, 660, 720, 780, 792, 840, 864, 900, 936, 960, 1008, 1020, 1080, 1140, 1152, 1200, 1224, 1260, 1296, 1320, 1368, 1380, 1440, 1500, 1512, 1560, 1584, 1620, 1656, 1680, 1710

Offset: 1

Michel Lagneau, Jun 07 2020

nonn

The subsequence of numbers of the form 2^i*3^j is 72, 144, 216, 288, 432, 576, 648, 864, 1152, 1296, ...
The corresponding number of quadruples of the sequence is 1, 1, 2, 2, 2, 2, 3, 3, 2, 6, 2, 4, 4, 2, 3, 4, 4, 3, 2, 10, ... (see the sequence A328204).
The set of divisors of a(n) contains at least one primitive quadruple.
Examples:
  The set of divisors of a(1) = 60 contains only one primitive quadruple: (3, 4, 5, 6).
  The set of divisors of a(10) = 360 contains two primitive quadruples: (1, 6, 8, 9) and (3, 4, 5, 6).
From Robert Israel, Jul 06 2020: (Start)
Every multiple of a member of the sequence is in the sequence.
The first member of the sequence not divisible by 6 is a(68) = 2380, which has the quadruple (7, 14, 17, 20).
The first odd member of the sequence is a(1230) = 43065, which has the quadruple (11, 15, 27, 29). (End)

			120 is in the sequence because the set of divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120} contains the quadruples {3, 4, 5, 6} and {6, 8, 10, 12}. The first quadruple is primitive.

Y. Perelman, Solutions to x^3 + y^3 + z^3 = u^3, Mathematics can be Fun, pp. 316-9 Mir Moscow 1985.

Robert Israel, Table of n, a(n) for n = 1..10000
Fred Richman, Sums of Three Cubes

Cf. A023042, A027750, A096545, A328204, A330893.

with(numtheory):
for n from 3 to 2000 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `,n):
    else fi:
   od:

nq[n_] := If[ Mod[n, 6]>0, 0, Block[{t, u, v, c = 0, d = Divisors[n], m}, m = Length@ d; Do[ t = d[[i]]^3 + d[[j]]^3; Do[u = t + d[[h]]^3; If[u > n^3, Break[]]; If[ Mod[n^3, u] == 0 && IntegerQ[v = u^(1/3)] && Mod[n, v] == 0, c++], {h, j+1, m - 1}], {i, m-3}, {j, i+1, m - 2}]; c]]; Select[ Range@ 1026, nq[#] > 0 &] (* program from Giovanni Resta adapted for the sequence. See A330893 *)

isok(n) = {my(d=divisors(n), m); if (#d > 3, for (i=1, #d-3, for (j=i+1, #d-2, for (k=j+1, #d-1, if (ispower(d[i]^3+d[j]^3+d[k]^3, 3, &m) && !(n%m), return (1));););););} \\ Michel Marcus, Nov 15 2020

A328204 Numbers of quadruples contained in the divisors of A328149(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 2, 6, 2, 4, 4, 2, 3, 4, 4, 3, 2, 10, 2, 2, 4, 6, 4, 2, 5, 4, 2, 10, 2, 5, 6, 2, 4, 6, 4, 2, 2, 14, 3, 4, 4, 4, 4, 2, 6, 1, 8, 2, 11, 2, 4, 6, 4, 4, 6, 4, 2, 4, 17, 2, 2, 4, 6, 4, 4, 1, 8, 4, 2, 12, 2, 9, 6, 2, 6, 4, 4, 4, 2, 18, 3, 2, 6

Offset: 1

Michel Lagneau, Jun 07 2020

nonn

A quadruple (x, y, z, w) of A328149 is a set of positive integers that satisfy x^3 + y^3 + z^3 = w^3.

			a(7) = 3 because the set of divisors of A328149(7) = 240: {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains the three quadruples {3, 4, 5, 6}, {6, 8, 10, 12} and {12, 16, 20, 24}. The first quadruple is primitive.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A027750, A096545, A328149, A330894.

with(numtheory):
for n from 3 to 3000 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `,it):
    else fi:
   od:

nq[n_] := If[Mod[n, 6] > 0, 0, Block[{t, u, v, c = 0, d = Divisors[n], m}, m = Length@ d; Do[t = d[[i]]^3 + d[[j]]^3; Do[u = t + d[[h]]^2; If[u > n^3, Break[]]; If[Mod[n^3, u] == 0 && IntegerQ[v = u^(1/3)] && Mod[n, v] == 0, c++], {h, j+1, m-1}], {i, m-3}, {j, i+1, m - 2}]; c]]; Select[Array[nq, 1638], # > 0 &] (* program from Giovanni Resta adapted for the sequence. See A330894 *)

isok(n) = {my(d=divisors(n), nb=0, m); if (#d > 3, for (i=1, #d-3, for (j=i+1, #d-2, for (k=j+1, #d-1, if (ispower(d[i]^3+d[j]^3+d[k]^3, 3, &m) && !(n%m), nb++););););); nb;}
lista(nn) = my(m); for (n=1, nn, if (m=isok(n), print1(m, ", "))); \\ Michel Marcus, Nov 15 2020

A385325 Numbers x such that there exist two integers y, z both >0 such that sigma(x)^3 = x^3 + y^3 + z^3.

Original entry on oeis.org

5, 6, 53, 58, 102, 118, 152, 168, 197, 214, 250, 258, 408, 426, 445, 476, 487, 491, 508, 672, 760, 783, 861, 885, 1182, 1204, 1242, 1299, 1305, 1350, 1615, 1890, 1988, 1992, 2040, 2082, 2190, 2465, 2519, 2679, 3105, 3144, 3213, 3276, 3292, 3432, 3994, 4035, 4210, 4256

Offset: 1

S. I. Dimitrov, Jun 25 2025

nonn

The numbers x, y and z form a sigma-cubic triple. See Dimitrov link.

If sigma(x)^3 = x^3 + y^3 + z^3 then sigma(x)^3 - x^3 = y^3 + z^3 = (y + z)*(y^2 - y*z + z^2) which enables comparing pairwise divisors of sigma(x)^3 - x^3 to see if sigma(x)^3 - x^3 is the sum of two cubes. - David A. Corneth, Jun 26 2025

			(3, 4, 5) is such a triple because sigma(5)^3 = 6^3 = 5^3 + 4^3 + 3^3.
6 is in the sequence as sigma(6)^3 = 6^3 + 8^3 + 10^3. - _David A. Corneth_, Jun 26 2025

David A. Corneth, PARI program
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A003325, A066784, A096545.

PARI
```
\\ See Corneth link
```

Data corrected by David A. Corneth, Jun 26 2025

A337098 Least k whose set of divisors contains exactly n quadruples (x, y, z, w) such that x^3 + y^3 + z^3 = w^3, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 432, 960, 360, 3840, 1728, 2592, 720, 1800, 2520, 161700, 1440, 6840, 9000, 2160, 2880, 168300, 5040, 41472, 5760, 1520820, 4320, 7200, 11520, 119700, 10080, 682080, 10800, 8640, 14400, 27360, 12960, 373248, 20160, 61560, 17280, 28800, 55440, 171000, 21600

Offset: 1

Michel Lagneau, Aug 15 2020

Observation: a(n) == 0 (mod 12).

Listing primitive tuples (w, x, y, z) enables to compute for some m how many such tuples are in its divisors using the lcm of such tuples. - David A. Corneth, Sep 26 2020

			a(3) = 240 because the set of the divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 quadruples {3, 4, 5, 6}, {6, 8, 10, 12} and {12, 16, 20, 24}. The first quadruple is primitive.

Y. Perelman, Solutions to x^3 + y^3 + z^3 = u^3, Mathematics can be Fun, pp. 316-9 Mir Moscow 1985.

David A. Corneth, Table of n, a(n) for n = 1..504
Fred Richman, Sums of Three Cubes

Cf. A027750, A095868, A095867, A096545, A096546, A328204, A328149, A331365.

with(numtheory):divisors(240);
for n from 1 to 52 do :
ii:=0:
for q from 6 by 6 to 10^8 while(ii=0) do:
   d:=divisors(q):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it = n
     then
     ii:=1: printf (`%d %d \n`,n,q):
     else
    fi:
od:
od:

With[{s = Array[Count[Subsets[Divisors[#], {4}]^3, ?(#1 + #2 + #3 == #4 & @@ # &)] &, 10^4]}, Rest@ Values[#][[1 ;; 1 + LengthWhile[Differences@ Keys@ #, # == 1 &] ]] &@ KeySort@ PositionIndex[s][[All, 1]]] (* _Michael De Vlieger, Sep 18 2020 *)

from itertools import combinations
from sympy import divisors
def A337098(n):
    k = 1
    while True:
        if n == sum(1 for x in combinations((d**3 for d in divisors(k)),4) if sum(x[:-1]) == x[-1]):
            return k
        k += 1 # Chai Wah Wu, Sep 25 2020

a(13)-a(22) from Chai Wah Wu, Sep 25 2020

More terms from David A. Corneth, Sep 26 2020

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cp's OEIS Frontend

A095867 Values y associated with A096545(n), sorted on z, then on y and finally on x.

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A095868 Values x associated with A096545(n), sorted on z, then on y and finally on x.

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A096546 Values w associated with A096545(n), sorted on z, then on y and finally on x.

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A328149 Numbers whose set of divisors contains a quadruple (x, y, z, w) satisfying x^3 + y^3 + z^3 = w^3.

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PARI

A328204 Numbers of quadruples contained in the divisors of A328149(n).

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PARI

A385325 Numbers x such that there exist two integers y, z both >0 such that sigma(x)^3 = x^3 + y^3 + z^3.

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A337098 Least k whose set of divisors contains exactly n quadruples (x, y, z, w) such that x^3 + y^3 + z^3 = w^3, or 0 if no such k exists.

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Maple