Turning 28: perfect, triangular and hexagonal¶

Well, after some time I return with posts that are finally related to something other than technology. The motivation for this publication is the particular birthday message that my brother left me, which says:

Happy 28th! 28 is a perfect number, literally. Because of that it is also hexagonal and triangular. It is true that it is the second perfect number, but it is also the last one you will celebrate. The next one is near 500. Could you find it?

Numerical sequences¶

It turns out that apart from the classification of traditional numbers that we know from basic education, such as integers, it is possible to assign names to different collections of numbers, associated with some curiosities (some useful, others not so much).

In particular, the numerical collections that interest me for this publication are numerical sequences. Unlike traditional ones, these are not generated from a mere definition based on characteristics that are obvious in the written form (for example, we easily distinguish real numbers by the decimal separator, fractionals by the division bar between integers, negatives by the negative symbol) but by a formulation, a mathematical expression that can represent their value or by some general pattern exhibited by the values (which is not associated with their writing or nature).

Now, let’s look at why turning 28 is perfect, triangular, and hexagonal.

Perfect numbers¶

Perfect numbers meet a rather particular property: a perfect number is that number whose positive proper divisors added together result in the mentioned number [oeis_perfect-en] [wiki_perfect-en].

This is somewhat uncommon, and the first perfect number is 6 and the second is 28. As the message indicated at the beginning mentions, the next one is around 500 (specifically, 496). Let’s see how we know it practically.

n = 6: We initially find the proper divisors of 6, which correspond to: 1, 2, and 3. Then: \(1+2+3=6\). Since it is equal, 6 is a perfect number.
n = 28: We find again the positive proper divisors: 1, 2, 4, 7, and 14. And we perform their sum: \(1+2+4+7+14=28\). Therefore, it is a perfect number.

By definition, 1 is not considered a perfect number; technically there is no sum to make as it is a single positive proper divisor. How do we know which is the next one? Well, that is the boring part if you want to know it on your own. Since there is no expression that generates it but a pattern associated with a characteristic of the number, the only option is to check number by number if it meets the mentioned condition.

Triangular numbers¶

Triangular numbers are geometric constructions from points, where the points are then counted, and that quantity indicates that the number belongs to this type. The same applies to the hexagonal number.

I explain: it is about using points to build geometry, in this case, an equilateral triangle. Thus, the numbers obtained are: 0, 1, 3, 6, 10, 15, 21, 28, and so on [wiki_tri-en].

Note that with 0 points we form a triangle whose sides have a length of 0 points (yes, curiously the definition includes 0). With one point, we form a triangle whose sides are 1 point. With 3 points we form a triangle whose sides are 2 points (since the corners are shared, that’s why it is 3 points in total), 6 points are triangles with sides of 3 points (3 vertices that are shared and 3 that are in the middle of the sides—one on each side—) and 10 has sides of 4 points (3 shared vertices, 2 points on each side and 1 interior point of the triangle since the previous triangle is accumulated).

.   .      .           .
   . .    . .         . .
         . . .       . . .
                    . . . .

From the mathematical point of view, triangular numbers are the partial sums of the sequence of positive integers \(T_{n} = \sum\limits_{i=1}^{n}i\), or equivalently the Newton binomials of the form [wiki_tri-en] [oeis_tri-en]

\[\begin{split}T_{n} = \begin{pmatrix}n+1 \\ 2\end{pmatrix} = \frac{n(n+1)}{2}\end{split}\]

Note that \(28 = 1+2+3+4+5+6+7\).

Hexagonal number¶

As I mentioned earlier, the hexagonal number can also be seen as a geometric arrangement of points forming a regular polygon (these numbers are generally called polygonal numbers). In this case the geometric arrangement corresponds to forming a regular hexagon with the points. Just like the triangular one, the points of the previous hexagon must be accumulated and reused (in this way there are two edges that share their points with the new hexagon) [wiki_hex-en].

.    ..         ...
    .  .       .  ..
     ..       .  .  .
               .  ..
                ...

Like the triangular ones, it also has a mathematical expression that can generate the sequence: \(n(2n-1)\) [wiki_hex-en] [oeis_hex-en].

What is the next one?¶

Well, at this time I’m already getting a bit sleepy, so I will give the answer in an update of the publication or in a new entry in which I can take the opportunity to explain some things about Python such as generators.

But what does programming have to do with it? Given the pattern exhibited by the numbers, it is not possible to simply substitute into an equation to find out the answer to the message’s question (but it can be quickly checked in the OEIS [oeis-en]), and it is necessary to make a code that verifies the condition for each number in an orderly manner.