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URL: https://oeis.org/A225908

⇱ A225908 - OEIS

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A225908
Numbers that are both a sum and a difference of two positive cubes.
5
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
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
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jun 21 2013
STATUS
approved