A225908 Numbers that are both a sum and a difference of two positive cubes.

91, 152, 189, 217, 513, 728, 1027, 1216, 1512, 1736, 2457, 3087, 4104, 4706, 4921, 4977, 5103, 5256, 5824, 5859, 6832, 7657, 8216, 8587, 9728, 10712, 11375, 12096, 12691, 13851, 13888, 14911, 15093, 15561, 16120, 16263, 19000, 19656, 21014, 23058, 23625, 24696

Offset

1,1

Comments

Solutions x to the equations x = a^3 + b^3 = c^3 - d^3 in positive integers.

The intersection of A003325 and A181123. See those sequences for additional comments, references, links and cross-refs.

Suggested by Shiraishi's solutions to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (transpose a term from the left side to the right side). See A023042 and A226903.

An infinite subsequence is (A226904(n)+1)^3 - A226904(n)^3.

References

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.

Wikipedia (French), Shiraishi Nagatada

Wikipedia (German), Shiraishi Nagatada

Index entries for sequences related to sums of cubes

Example

3^3 + 4^3 + 5^3 = 6^3, so 3^3 + 4^3 = 91 and 3^3 + 5^3 = 152 and 4^3 + 5^3 = 189 are members.

Mathematica

nn = 3*10^4; t1 = Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]; p = 3; t2 = Union[Reap[Do[n = i^p - j^p; If[n <= nn, Sow[n]], {i, Ceiling[(nn/p)^(1/(p - 1))]}, {j, i}]][[2, 1]]]; Intersection[t1, t2] (* T. D. Noe, Jun 21 2013 *)

Crossrefs

Cf. A003325, A023042, A181123, A226903, A226904.

Sequence in context: A045934 A051347 A293647 * A159961 A113530 A119148

Adjacent sequences: A225905 A225906 A225907 * A225909 A225910 A225911

Keyword

nonn

Author

Jonathan Sondow, Jun 21 2013

Status

approved