cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A277857 Numbers that are the sum of 2 squares with a unique partition and also the sum of 3 nonnegative cubes with a unique partition.

Original entry on oeis.org

1, 2, 8, 9, 10, 16, 17, 29, 36, 64, 72, 73, 80, 81, 128, 136, 153, 160, 197, 218, 232, 244, 277, 281, 288, 314, 349, 397, 405, 433, 466, 468, 512, 514, 521, 557, 576, 577, 584, 586, 593, 637, 640, 648, 701, 738, 757, 794, 801, 853, 857, 881, 882, 953, 980, 1024, 1028, 1088, 1152, 1217, 1224, 1249, 1268, 1280, 1332, 1341, 1396
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 02 2016

Keywords

Comments

Primes in this sequence are 2, 17, 29, 73, 197, 277, 281, 349, 397, 433, 521, 557, 577, 593, 701, 757, 853, 857, 881, 953, ... (subsequence of A002313).

Examples

			a(1) = 1 because 1 = 0^2 + 1^2 and 1 = 0^3 + 0^3 + 1^3;
a(2) = 2 because 2 = 1^2 + 1^2 and 2 = 0^3 + 1^3 + 1^3;
a(3) = 8 because 8 = 2^2 + 2^2 and 8 = 0^3 + 0^3 + 2^3;
a(4) = 9 because 9 = 0^2 + 3^2 and 9 = 0^3 + 1^3 + 2^3;
a(5) = 10 because 10 = 1^2 + 3^2 and 10 = 1^3 + 1^3 + 2^3, etc.

Links

Crossrefs

Cf. A001481, A002313, A003072, A004825, A125022.

Programs

  • Mathematica
    Select[Range[1400], Length[PowersRepresentations[#1, 2, 2]] == 1 && Length[PowersRepresentations[#1, 3, 3]] == 1 & ]

A302360 Numbers that are the sum of 3 cubes > 1.

Original entry on oeis.org

24, 43, 62, 80, 81, 99, 118, 136, 141, 155, 160, 179, 192, 197, 216, 232, 251, 253, 258, 270, 277, 288, 307, 314, 344, 349, 359, 368, 375, 378, 397, 405, 415, 434, 440, 459, 466, 471, 476, 495, 496, 528, 532, 547, 557, 566, 567, 584, 586, 593, 603, 623, 640, 645, 648, 664, 684, 694, 701, 713, 736, 745, 750
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			118 is in the sequence because 118 = 3^3 + 3^3 + 4^3.

Links

Crossrefs

Cf. A003072, A004825, A025456, A258865, A294073.

Programs

  • Mathematica
    max = 750; f[x_] := Sum[x^(k^3), {k, 2, 10}]^3; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]]
Total/@Tuples[Range[2,10]^3,3]//Union (* Harvey P. Dale, May 26 2019 *)

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

Views

Author

XU Pingya, Jul 10 2020

Keywords

Comments

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.

Examples

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

References

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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

XU Pingya, Jul 17 2020

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

XU Pingya, Jul 12 2020

Keywords

Comments

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.

Examples

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

References

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

Crossrefs

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

Programs

  • Mathematica
    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 2*x^3 + 2*y^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

Views

Author

XU Pingya, Oct 18 2020

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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, #] &]
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.