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

Author

Jonathan Sondow, Jun 21 2013

Keywords

• nonn

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.

Examples

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

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

Crossrefs

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

Programs

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 *)

Formula

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