A004826 -id:A004826 - cp's OEIS frontend

A351338 Least nonnegative integer m such that n = x^3 + y^3 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

0, 0, 0, 5, 11, 4, 1, 1, 0, 0, 3, 2, 2, 35, 1, 1, 0, 7, 2, 2, 2, 12, 14, 10, 4, 1, 1, 0, 0, 3, 3, 44, 22, 1, 1, 0, 3, 3, 2, 8, 8, 127, 4, 7, 3, 2, 2, 8, 2, 2, 97, 7, 1, 1, 0, 2, 2, 2, 17, 13, 4, 4, 1, 1, 0, 0, 6, 20, 4, 4, 1, 1, 0, 15, 3, 2, 53, 22, 7, 3, 4, 6, 2, 2, 5, 14, 139, 4, 4, 1, 1, 0, 5, 3, 5, 22, 4, 3, 3, 3, 3

Offset: 0

Zhi-Wei Sun, Feb 08 2022

nonn

Conjecture: a(n) exists for any n >= 0. Equivalently, each integer can be written as x^3 + y^3 - (z^3 + w^3) with x,y,z,w nonnegative integers.

This is stronger than Sierpinski's conjecture which states that any integer is a sum of four integer cubes.

			a(41) = 127 with 41 = 41^3 + 128^3 - 49^3 -127^3.
a(130) = 143 with 130 = 37^3 + 169^3 - 125^3 - 143^3.
a(4756) = 533 with 4756 = 265^3 + 538^3 - 284^3 - 533^3.
a(5134) = 389 with 5134 = 19^3 + 418^3 - 242^3 - 389^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000578, A004826, A004999, A351306, A351321.

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

A001245 Numbers that are the sum of 4 cubes in more than 1 way.

Original entry on oeis.org

81, 126, 128, 216, 217, 219, 224, 243, 251, 252, 259, 278, 280, 315, 341, 343, 344, 345, 352, 371, 376, 378, 405, 408, 432, 434, 467, 469, 496, 522, 540, 559, 560, 567, 584, 593, 594, 648, 687, 702, 728, 729, 730, 737, 756, 758, 763, 765, 783, 793, 802

Offset: 1

N. J. A. Sloane

nonn

G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001235, A004826.

Transpose[Select[Table[{i,PowersRepresentations[i,4,3]}, {i,2000}], Length[#[[2]]]>1&]][[1]] (* Harvey P. Dale, Dec 11 2010 *)

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, #] &]

A336166 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2.

Original entry on oeis.org

0, 1, -3, 4, 9, -12, 16, 25, -27, -35, 36, 37, -48, 49, -59, 64, -75, 81, 100, -108, 121, 144, -147, -159, 169, 172, -192, 196, 225, -227, -243, -255, 256, 261, -287, 289, -300, -311, 324, -335, 361, -363, 373, 400, -432, 441, 484, -507, 529, 568, 576, -588

Offset: 1

XU Pingya, Jul 10 2020

sign

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

Segre shows that 1-(9/2)*A000578(2n), (-3)*A000290(n), and A016754(n) are terms of the sequence.

			(-5)^3 + (-11)^3 + 2 * 9^3 = 2, 9 is a term.
(25)^3 + (-23)^3 + 2 * (-12)^3 = 2, -12 is a term.

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

Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Cf. A000290, A000578, A004825, A004826, A016754, A028387, A050791, A130472, A195006.

t1 = Union[Plus@@@Tuples[Range[-11643, 11643]^3, 2]];
t2 = Table[2 - 2z^3, {z, -588, 588}];
t = Select[t1, MemberQ[t2, #] &];
u = ((2 - t)/2)^(1/3) /. (-1)^(1/3) -> (-1);
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 1176}];
Select[v, MemberQ[u, #] &]

A336226 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 1458.

Original entry on oeis.org

1, -3, 4, 9, -10, -12, 16, 21, 25, 37, -47, -48, 49, 64, -75, -87, 88, 100, 105, 121, 134, -147, 169, 172, -192, 196, -241, -243, 256, 289, -300, 361, -363, 400, 443, 484, -507, 529, 541, -588, 625, 676, -699, 732, -759, -768, 777, 784, 841, -867, 897, 961

Offset: 1

XU Pingya, Jul 17 2020

sign

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).
(11 + 3*n - 9*n^2)^3 + (11 + 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 1)^6 = 1458, the numbers of the form (3*n + 1)^2 are terms of the sequence.
(11 - 3*n - 9*n^2)^3 + (11 - 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 2)^6 = 1458, the numbers of the form (3*n + 2)^2 are also terms of the sequence.
Thus, A001651(n)^2 are terms of the sequence. There is an infinity of nontrivial solutions to the equation.

			5^3 + 11^3 + 2 * 1^3 = 1458, 1 is a term.
(-1)^3 + (11)^3 + 2 * (4)^3 = 1458, 4 is a term.

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

Cf. A000290, A000578, A001651, A003215, A004825, A004826, A050791, A130472, A195006.

Clear[t]
t = {};
Do[y = (1458 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -980, 980}, {x, -25319, 25319}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 2000}];
Select[v, MemberQ[u, #] &]

A336230 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 128.

Original entry on oeis.org

1, 4, 9, 25, 49, 81, 121, 169, -224, 225, 289, 361, -383, 441, 504, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, -2219, 2401, 2601, -2687, 2809, 3025, 3249, 3481, -3680, 3721, 3969, 4225, -4283, 4417, 4489, 4761, 5041, 5329, -5459

Offset: 1

XU Pingya, Jul 12 2020

sign

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

(5 - 4*n^2)^3 + (5 - 4*(n + 1)^2)^3 + 2*(2*n + 1)^6 = 128. A000290(2*n + 1) are terms of the sequence, i.e., there is an infinity of nontrivial solutions to the equation.

			1^3 + 5^3 + 2 * 1^3 = 128, 1 is a term.
(-11)^3 + (-31)^3 + 2 * (25)^3 = 128, 25 is a term.

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

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

Clear[t]
t = {};
Do[y = (128 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -4761, 4761}, {x, -11550, 11550}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 9523}];
Select[v, MemberQ[u, #] &]

A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2x^3 + 2y^3 + z^3 = 1.

Original entry on oeis.org

-1, 1, -5, 11, -17, 19, 29, -31, -37, -61, 79, -85, 113, -127, -143, 145, -209, 305, 361, -485, 487, 545, 647, 667, 811, -1091, -1151, 1153, -1235, -1429, -1525, 1597, 1699, -1793, -2249, 2251, -2533, 2627, -2677, 2977, -2981, 3089, -3295, 3739, -3887, 3889

Offset: 1

XU Pingya, Oct 18 2020

sign

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

When x = (3*c)*t - (9*a)*t^4, y = (9*a)*t^4, z = c - (9*a)*t^3; a*x^3 + a*y^3 + c*z^3 = c^4. Let a = 2, c = 1, then 1 - 18*n^3 and 1 + 18*n^3 are terms of the sequence. Also, -A337928 and A337929 are subsequences.

			2*25^3 + 2*(-64)^3 + 79^3 = 2*164^3 + 2*(-167)^3 + 79^3 = 1, 79 is a term.

Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Cf. A000045, A000290, A000578, A003215, A004825, A004826, A047201, A050791, A130472, A195006, A337928, A337929.

Clear[t]
t = {};
Do[y = ((1 - 2x^3 - z^3)/2)^(1/3) /. (-1)^(1/3) -> -1;
 If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -4000, 4000}, {x, -Round[(Abs[1 + z^3]/6)^(1/2)], Round[(Abs[1 + z^3]/6)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 8001}];
Select[v, MemberQ[u, #] &]

cp's OEIS Frontend

A351338 Least nonnegative integer m such that n = x^3 + y^3 - (z^3 + m^3) for some nonnegative integers x,y,z with z <= m.

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A001245 Numbers that are the sum of 4 cubes in more than 1 way.

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A336449 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*4^6.

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A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*5^6.

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A336166 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2.

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A336226 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 1458.

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A336230 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 128.

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A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1.

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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.

A338239 Values z of primitive solutions (x, y, z) to the Diophantine equation 2x^3 + 2y^3 + z^3 = 1.