A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.

91, 217, 1027, 4921, 8587, 14911, 31519, 39331, 106597, 117019, 136747, 185257, 195841, 265519, 281827, 616987, 636181, 684019, 712969, 724717, 736561, 955981, 1200169, 1352737, 1405621, 1771777, 2481571, 2756167, 2937331, 4251871, 4996171, 5262901

Offset

1,1

Comments

Solutions x to the equations x = a^3 + b^3 = (c+1)^3 - c^3 in positive integers. The values of c are A226902.

The intersection of A003325 and A003215.

Subsequence of A225908 = numbers that are both a sum and a difference of two positive cubes.

Shiraishi's solution to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (see A023042 and A226903) shows that the sequence is infinite.

References

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..5000

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

Formula

a(n) = (A226902(n)+1)^3 - A226902(n)^3.

Example

3^3 + 4^3 = 6^3 - 5^3 = 91, so 91 is a member.

Mathematica

Select[#[[2]]-#[[1]]&/@Partition[Range[5000]^3, 2, 1], Count[ IntegerPartitions[ #, {2}], _?(AllTrue[Surd[#, 3], IntegerQ]&)]>0&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 07 2018 *)

Crossrefs

Cf. A003215, A003325, A023042, A181123, A225908, A226902, A226903.

Sequence in context: A260064 A207077 A293648 * A051973 A290812 A000864

Adjacent sequences: A225906 A225907 A225908 * A225910 A225911 A225912

Keyword

nonn

Author

Jonathan Sondow, Jun 21 2013

Status

approved