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.

Showing 1-3 of 3 results.

A051347 Numbers that are the sum of two (possibly negative) cubes in at least 2 ways.

Original entry on oeis.org

91, 152, 189, 217, 513, 721, 728, 999, 1027, 1216, 1512, 1729, 1736, 2457, 3087, 3367, 4104, 4706, 4921, 4977, 5103, 5256, 5768, 5824, 5859, 6832, 7657, 7992, 8216, 8587, 8911, 9728, 9919, 10621, 10712, 11375, 12096, 12663, 12691, 12824, 13832, 13851
Offset: 1

Views

Author

Colin Mallows

Keywords

Examples

			91 = 3^3 + 4^3 = (-5)^3 + 6^3; 152 = 3^3 + 5^3 = (-4)^3 + 6^3; 189 = 4^3 + 5^3 = (-3)^3 + 6^3; ...

Links

Crossrefs

Cf. A051383, A051384.

Programs

  • Mathematica
    ok[n_] := If[Length[PowersRepresentations[n, 2, 3]] >= 2, True, r = Reduce[n == x^3 + y^3, {x, y}, Integers]; If[r === False, False, Length[Union[Sort /@ ({x, y} /. {ToRules[r]})]] >= 2]]; Select[Range[13860], If[ok[#], Print[#]; True, False] &] (* Jean-François Alcover, Apr 11 2011 *)
  • PARI
    is(n)=#thue(thueinit(z^3+1),n)>=2 \\ Ralf Stephan, Oct 18 2013

Extensions

Extended by Ray Chandler, Jan 30 2009

A293650 Sum of two (possibly negative) cubes in at least 3 ways (primitive solutions).

Original entry on oeis.org

728, 3367, 4104, 5859, 46683, 65728, 68913, 101528, 124488, 134379, 152551, 155736, 165464, 168112, 184464, 195841, 205352, 289224, 333944, 342657, 402597, 439101, 622232, 625177, 684019, 754299, 757701, 842751, 845208, 1009736, 1016496, 1062936, 1073375
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Primitive means that the 6 summands are coprime. Not every term is the sum of two coprime cubes. a(1) = A047696(3).

Examples

			4104 = 18^3 - 12^3 = 16^3 + 2^3 = 15^3 + 9^3, and 18, -12, 16, 2, 15, 9 are coprime, so 4104 is in the sequence.

Links

Crossrefs

Cf. A051383 (all solutions); A003825 (positive cubes); A293651 (only coprime); A293645, A293647

A051384 Sum of two (possibly negative) cubes in at least 4 ways.

Original entry on oeis.org

2741256, 4118877, 6017193, 6742008, 9016488, 16776487, 21930048, 28699272, 32951016, 36875384, 42549416, 48137544, 48275136, 52324993, 53936064, 70957971, 72131904, 74013912, 87539319, 94287375, 102977784, 105651000, 111209679, 119824488, 122262264, 124454421, 134211896
Offset: 1

Views

Author

Colin Mallows

Keywords

Comments

This sequence is infinite, since if n is in the sequence so is n*k^3 for all k > 0; thus a(n) << n^3. - Charles R Greathouse IV, Nov 29 2014

Examples

			42549416 = 348^3+74^3 = 282^3+272^3 = (-2662)^3+2664^3 = (-475)^3+531^3, so 42549416 is in the sequence. (Silverman)

Links

Crossrefs

Cf. A051347, A051383.

Programs

  • PARI
    T=thueinit('z^3+1);is(n)=my(v=thue(T, n)); #v>6 && #select(u->u[1]<=u[2],v)>3 \\ Charles R Greathouse IV, Nov 29 2014

Formula

a(1) = A047696(4). - R. J. Mathar, Aug 28 2025

Extensions

a(6)-a(22) from Donovan Johnson, Apr 17 2010
Missing terms 42549416, 48275136, 94287375, 111209679, 124454421 added by Rosalie Fay, Oct 13 2017
Showing 1-3 of 3 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.