28260: An Interesting Number

Sometimes it's difficult to find many interesting properties for numbers greater than 28000. However, 28260 is definitely not one of those sorts of numbers. It has a plethora of interesting properties and in this post I'll list some of them. Firstly though, let's list its prime factorisation:$$ \textbf{28260} = 2^2 \times 3^2 \times 5 \times 157$$PROPERTY 1: sum of two squares

As can be seen, the number is a product of a power of 2 (\( 2^2\)), a 4\(k\)+1 prime raised to an even power (\(3^2\)) and two 4\(k\)+1 primes (5 and 157). This means that it can be expressed as a sum of two squares in two different ways, viz.:$$ \begin{align} \textbf{28260} &= 6^2+168^2\\ \textbf{28260} &= 96^2+138^2 \end{align} $$PROPERTY 2: a, b, c, d number

It is what I've termed an \(a, b, c, d\) number because its digits can be rearranged to form three different numbers with the property that \(a+b+c=d\) where all four numbers share identical digits. In this case, there are two possible arrangements:$$ \begin{align} 26082 + 26280 + \textbf{28260} &= 80622\\ 26280 + 28062 + \textbf{28260} &= 82602 \end{align} $$PROPERTY 3: d-powerful number

It is digitally powerful (or \(d\)-powerful) because it can be expressed as a sum of positive powers of its digits. Here we have:$$ \textbf{28260} = 2^{14} + 8^4 + 2^2 + 6^5 + 0$$Of additional interest is that this number is the beginning of chain of ten consecutive numbers with this property. See Figure 1.

Figure 1

PROPERTY 4: Ulam number

It is an Ulam number. The Ulam sequence is defined by \(U_1=1\), \(U_2=2\)  and, for \(k>2\), \(U_k\) is the smallest integer that can be written in exactly one way as \(U_i+U_j\) with \(i<j<k\). Here we have:$$ \textbf{28260}=3 + 28257$$PROPERTY 5: gapful number

It a gapful number defined as a number of at least 3 digits that is divisible by the number formed by its first and last digits. Here the first and last digits form the number 20 and 20 is indeed a divisor:$$ \frac{ \textbf{28260}}{20} = 1413$$PROPERTY 6: untouchable number

It is an untouchable number defined as a number \(n\) that is not the sum of the proper divisors of any number \(k\). In other words:$$ \textbf{28260} \neq \sigma(k)-k \text{ for any }k $$PROPERTY 7: inconsummate number

It is an inconsummate number defined as a number \(n\) for which there is no number $k$ such that \(k\) divided by its sum of digits (SOD) gives $n$. Thus we have:$$ \frac{k}{\text{SOD}(k)}\neq \textbf{28260} \text{ for any } k$$PROPERTY 8: tau number

is a tau number since it is divisible by its number of divisors. Here there are 36 divisors and we have:$$ \frac{\textbf{28260}}{36}=785$$PROPERTY 9: Harshad number

It is a Harshad number defined as number that is divisible by the sum of its digits. Here we have a sum of digits of 18 and:$$ \frac{\textbf{28260}}{18}=1570$$PROPERTY 10: zeroes of the Mertens function

It is a number where the Mertens function has a value of zero. It forms a pair of consecutive numbers with 28259.

I examined the Möbius and Mertens functions in a post titled The Möbius Function and Mertens Function on January 25th 2020. In number theory, we define the Mertens function as:$$M(n) = \sum_{1\le k \le n} \mu(k)$$where \( \mu (k)\) is the Möbius function. For any positive integer n, \(μ(n)\) has values in {−1, 0, 1} depending on the factorisation of \(n\) into prime factors:$$\mu(n) = \begin{cases} 1 & \quad \text{if } n \text{ is square-free + integer with even number of prime factors}\\ -1 & \quad \text{if } n \text{ is square-free + integer with odd number of prime factors}\\ 0 & \quad \text{if } n \text{ has a squared prime factor} \end{cases}$$Figure 2 shows the situation:

Figure 2

There are many other properties that the number has but this covers most of the more interesting.