The Curious 6

6 is such a very intriguing number. It is perhaps best known for being the first perfect number, which was noted by Euclid in 300 BCE. A perfect number is a number whose factors sum to the number itself. In the case of 6, $1+2+3=6$ . Perfect numbers are rare beasts indeed; the fact the the 6th perfect number is 8,589,869,056 should demonstrate that!

6 is also a triangle number, which means we can arrange a set of 6 objects into a triangle. It is also the first composite number without repeated prime factors, $6=2 \cot 3$.

Triangle numbers are generated by:

$T_{n} = \frac{(n)(n+1)}{2}$ explicitly or by $T_{n} = T_{n-1} + n$ recursively.

So $T_{3} = 6$, that is, 6 is the triangle number with generator 3. What is the triangle number with generator 6? Well,

$T_{6} = \frac{(6)(6+1)}{2} = \frac{56}{2} = 28$.

This may seem unremarkable, until we realize 28 is the next consecutive perfect number after 6! Adding up the factors of 28:

$T_{6} = 28 = 1 + 2 + 4 + 7 + 14$.

I do not know if 6 being the triangle generator of 28 is related to them both being perfect. All known perfect numbers are also triangle numbers, but this does not mean they generate one another. For instance, the next perfect number after 28 is 496, which is $T_{31}$ not $T_{28}$. It is only 6 that manages to do both.

There is one other curious property of 6 that had me thinking, and perhaps offers insight into relating these things more elegantly. 6 is the first number base with a number less than itself (4) that does not divide into the base itself but does divide perfectly into a power of the base (in this case $6^{2} = 36$ and $4 | 36$ ). The | symbol there means "divides into", but is not used much outside of modular arithmetic. Thus, 4 is neither coprime with 6, nor a divisor of it. If nothing else, it indicates that 36 has a lot of factors!

I'm wondering if any fellow 6-lovers have any more insight on this.