A095867 -id:A095867 - cp's OEIS frontend

A096545 Ordered z such that, for 0

5, 8, 17, 18, 21, 22, 27, 33, 37, 37, 40, 41, 44, 49, 53, 54, 57, 61, 64, 65, 66, 69, 69, 70, 72, 74, 75, 78, 79, 79, 79, 84, 85, 86, 86, 87, 89, 90, 92, 96, 97, 97, 97, 99, 101, 102, 102, 104, 105, 108, 114, 116, 118, 121, 122, 123, 124, 124, 128, 131, 136, 136, 137

Offset: 1

Lekraj Beedassy, Jun 25 2004

nonn

For corresponding values w see A096546.

			21 and 22, for instance, are terms because we have: 18^3 + 19^3 + 21^3 = 28^3 and 4^3 + 17^3 + 22^3 = 25^3.

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..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]&;
Sort[xyzw[[All, 3]]] (* Jean-François Alcover, Mar 06 2020 *)

Edited, corrected and extended by Ray Chandler, Jun 28 2004

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

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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A096545 Ordered z such that, for 0

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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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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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