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

Author

Jonathan Sondow, Jun 21 2013

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

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

Programs

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 *)