Sums of Cubes

 I've written about cubic numbers before, specifically in posts titled:

• Cubic Numbers from the 11th January 2016 in which I discuss turning \(29^3\) days old
• Cuban Primes from the 11th July 2016 in which I discuss primes that are the difference of two consecutive cubes
• Sums of Cubes and Squares of Sums from the 31st July 2018 in which I discuss with the property that the sum of the cubes of their digits is equal to the square of the sums of their digits
• Platonic Numbers from the 20th November 2018 in which I discuss cubes or cubic numbers as being one type of platonic number along with tetrahedral, octahedral, icosahedral and dodecahedral numbers.
• 42 is the new 33 from the 1st April 2019 in which I report on the discovery that 33 is expressible as a sum of three signed cubes and that only 42 remains to be cracked.
• 42 from the 9th September 2019 in which I report on the discovery is expressible as a sum of three signed cubes and that a new way of expressing 3 as a sum of three signed cubes was discovered
• The Original Taxi Cab Number in a New Light from the 21st December 2019 in which I discuss 1729 and centred cube numbers

In this post, I'll examine 26132 that has the interesting property that it can be expressed as a sum of six cubes in three possible ways. It is a member of OEIS A048931

A048931

Numbers that are the sum of 6 positive cubes in exactly 3 ways.

What's remarkable about 26132 is that it's the last such number with this property. Beginning with 221, the sequence has 1141 members and 26132 is the last member. It is expressible as the sum of the following cubic numbers: [8, 2744, 2744, 3375, 8000, 9261], [64, 1331, 2744, 4913, 4913, 12167] and [512, 512, 2197, 2744, 8000, 12167].

From WolframMathWorld we learn that:

• 23 and 239 are the only integers requiring nine positive cubes
• only 15 integers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 

The same source also provides the following table that gives the first few numbers which require at least N = 1, 2, 3, ..., 9 (i.e., N or more) positive cubes to represent them as a sum.

	OEIS	numbers
1	A000578	1, 8, 27, 64, 125, 216, 343, 512, ...
2	A003325	2, 9, 16, 28, 35, 54, 65, 72, 91, ...
3	A047702	3, 10, 17, 24, 29, 36, 43, 55, 62, ...
4	A047703	4, 11, 18, 25, 30, 32, 37, 44, 51, ...
5	A047704	5, 12, 19, 26, 31, 33, 38, 40, 45, ...
6	A046040	6, 13, 20, 34, 39, 41, 46, 48, 53, ...
7	A018890	7, 14, 21, 42, 47, 49, 61, 77, ...
8	A018889	15, 22, 50, 114, 167, 175, 186, ...
9	A018888	23, 239

Again, the same source provides the following table that gives the numbers which can be represented in exactly \(W\) different ways as a sum of \(N\) positive cubes.

		OEIS	numbers
1	0	A007412	2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ...
1	1	A000578	1, 8, 27, 64, 125, 216, 343, 512, ...
2	0	A057903	1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...
2	1		2, 9, 16, 28, 35, 54, 65, 72, 91, ...
2	2	A018850	1729, 4104, 13832, 20683, 32832, ...
2	3	A003825	87539319, 119824488, 143604279, ...
2	4	A003826	6963472309248, 12625136269928, ...
2	5		48988659276962496, ...
2	6		8230545258248091551205888, ...
3	0	A057904	1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...
3	1	A025395	3, 10, 17, 24, 29, 36, 43, 55, 62, ...
3	2		251, ...
4	0	A057905	1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...
4	1	A025403	4, 11, 18, 25, 30, 32, 37, 44, 51, ...
4	2	A025404	219, 252, 259, 278, 315, 376, 467, ...
5	0	A057906	1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...
5	1	A048926	5, 12, 19, 26, 31, 33, 38, 40, 45, ...
5	2	A048927	157, 220, 227, 246, 253, 260, 267, ...
6	0	A057907	1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, ...
6	1	A048929	6, 13, 20, 27, 32, 34, 39, 41, 46, ...
6	2	A048930	158, 165, 184, 221, 228, 235, 247, ...
6	3	A048931	221, 254, 369, 411, 443, 469, 495, ...

In the final row of the above table, we see the sequence to which 26132 belongs. There's a lot more of course that could be said about numbers formed from the sum of cubes but what we do know is that, after 26132, there are no more numbers that can be formed from six cubes in only three ways. How we know this, I don't know!