Celebrating 27720

It's not often that numbers as large as 27720 attract 632 entries in the Online Encycopedia of Integer Sequences (OEIS). By contrast, 27719 attracts 31 entries and 27721 attracts 27 entries. So what's so special about 27720?

Well, it has lots of interesting properties. Let's look at some of them. 

$\textbf{PROPERTY 1}$

The very first entry in the database is OEIS A002182:

A002182  Highly composite numbers: numbers $n$ where d($n$), the number of divisors of $n$  increases to a record.

The initial record holders, up to 40000, are as follows where we see 27720 is a member:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720

Table 1 shows the details:

Table 1: permalink

$\textbf{PROPERTY 2}$

In a similar vein is OEIS A004394 where 27720 also features:

A004394    superabundant numbers: $n$ such that $\sigma(n)/n > \sigma(m)/m$ for all $m < n$, $\sigma(n)$ being A000203(n), the sum of the divisors of $n$.

The initial members are as follows with most being the same as for OEIS A002182:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720

Table 2 shows the details:

Table 2: permalink

$\textbf{PROPERTY 3}$

Another interesting property of the number arises from its appearance in the denominator of the progressive sum of the harmonic numbers. These denominators constitute OEIS A002805.

A002805    denominators of harmonic numbers $\text{H}(n) =\displaystyle \sum_{i=1} ^n \dfrac{1}{i}$

The first terms in the sequence are 1, 2, 6, 12, 60, 20, 140, 280, 2520, 2520, 27720, 27720. Table 3 shows the details.

Table 3: permalink

$\textbf{PROPERTY 4}$

The number also arises from a quite simple recurrence relation:

A052542     $\text{a}(n) = 2 \times \text{a}(n-1) + \text{a}(n-2), \text{ with } \text{a}(0) = 1, \text{a}(1) = 2, \text{a}(2) = 4$

The initial members of the sequence are 1, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720 (permalink).

$\textbf{PROPERTY 5}$

Since $27720 = 2^3 \times 3^2 \times 5 \times 7 \times 11$, it is 12 times the product of the primorial number $2310 = 2 \times 3 \times 5 \times 7 \times 11$ and this qualifies it for membership in OEIS A129912 because 12 is itself a product of primorials viz. 2 x 6.

A129912 numbers that are products of distinct primorial numbers (see A002110).

The initial members of the sequence (with the primorials themselves included) can be generated using this permalink:

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800

Initially I misinterpreted this sequence as meaning numbers that are multiples of primorials but this is not the case. Instead the multiples themselves must be products of primorials and this is far more restrictive.