Integration tools: Vaktorions wekobions monchosin numbers and related matters

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The four numbers $1, 2, 145 \textrm{ and } 40585$ have the following peculiar property:

$1!=1, 2!=2, 1!+4!+5!=145 \textrm{ and } 4!+0!+5!+8!+5! =40585$ .

Numbers that have this property – that the sum of the factorials of their digits equals the number themselves – are called factorions.

It turns out these 4 numbers are the only whole number factorions, a point we will turn to below.

The numbers $1 \textrm{ and } 3435$ have the peculiar property that $1^1=1 \textrm{ and } 3^3+4^4+3^3+5^5$ .

Numbers $n=d_k\ldots d_1d_0$ (base 10 digits) with the property that $d_k^{d_k}+\ldots d_0^{d_0}=n$ are called Münchausen numbers, and dependent on how we define $0^0$ – either as $1 \textrm{ or } 0$ – there are 2 or 3 of them.

Are there any positive integers $n=d_k\ldots d_1d_0$ (base 10 digits) with the property that $d_k^3+\ldots d_0^3=n$ ?

Yes: 1, 153, 370, 371, 407 all have this property – we call them cubions – and they are the only positive integer cubions.

The three sets of examples above can all be encapsulated as instances of the following general notion.

Let $F$ be a function defined on non-negative integers and taking non-negative integer values.

We call a non-negative integer $n=d_k\ldots d_1d_0$ (base 10 digits) an F-ion if $F(d_k)+\ldots F(d_0)=n$

Factorions correspond to the function $f(n)=n!$ , Münchausen numbers to the function $f(n)=n^n$ (with an appropriate choice of $0^0$ ), and cubions to the function $F(n)=n^3$ .

Notation: for $n=d_k\ldots d_1d_0$ (base 10 digits) we will denote $F(d_k)+\ldots F(d_0)=n$ by $n^{[F]}$

We will see that for any function $F$ the size of F-ions is bounded by an number that we can calculate numerically.

This allows us to conclude that for such $F$ the number of F-ions is finite.

Moreover, because we can numerically calculate the bound, we can do a computational search up to that bound to see if we have found all F-ions.

For a positive integer $n$ the number of (base 10) digits of $n$ is the floor of $\log_{10}(n)+1$ , the largest integer less than or equal to $\log_{10}(n)+1$ .

This means that the number of digits of $n$ is less than $\log_{10}(n)+1$ .

If the base 10 digits of the positive integer $n$ are $d_k,\ldots , d_1, d_0$ then:

$0\leq d_i \leq 9$ for $0\leq i\leq k$ The number of digits of $n$ is $k+1 \leq \log_{10}(n)+1$

Let $A\textrm{ denote }\max\{F(0),\ldots , F(9)\}$

Then we have: $n^{[F]}=F(d_k)+\ldots +F(d_1)+F(d_0) \leq (k+1) A \leq A(\log_{10}(n)+1)$

This says that $n^{[F]}$ cannot grow too big in terms of the number of digits of $n$ .

We look at the behavior of the function $G(x):=\frac{x}{\log_{10}(x)}$ :

Graph of x/log10(x) for x>1

The derivative of $G(x)$ is $\frac{dG(x)}{dx}=\log(10)\frac{\log(x)-1}{\log^2(x)}$ and this is positive for $x>e$ so for $n\geq 3$ the function $n\to \frac{n}{\log_{10}(n)+1}$ is strictly increasing.

Therefore there is a smallest non-negative integer $n^*$ for which $\frac{n^*}{\log_{10}(n^*)+1}>A$ .

For $n\geq n^*$ we have $\frac{n}{\log_{10}(n)+1}\geq \frac{n^*}{\log_{10}(n^*)+1}>A$ and so $n\geq A (\log_{10}(n)+1)$ .

So, for $n\geq n^*$ we cannot have $n=n^{[F]}$ , so showing that the number of F-ions is finite.

The least non-negative integer $n^*$ for which $\frac{n^*}{\log_{10}(n^*)+1}>A$ is the ceiling of the unique solution $x^*$ to $\frac{x^*}{\log_{10}(x)+1}=A$ .

There are a number of ways to estimate the solution $x^*$ to $\frac{x^*}{\log_{10}(x)+1}=A$ .

One way, using Mathematica, is:

> FindRoot[x/Log[10, x] == A, {x, substitute guesstimate for x*}]

For example, to get an upper bound for cubions, where $F(n)=n^3$ we first calculate $A=\max\{F(0),\dots , F(9)\}=9^3=729$ and guesstimate from a graph of $\frac{x}{log_{10}(x)+1}$ that $x\approx 3000$ :

The Mathematica code:

FindRoot[x/Log[10, x] == 9^3, {x, 3000}]

gives $x^*=2473.76$ so $n^*=2474$ .

The positive cubions we found were 1, 153, 370, 371, 407 and an easy check up to $n^*=2474$ shows there are no others.

A squarion is a positive integer $n$ for which $n=n^{[F]}$ where $F(n)=n^2$ .

There are no positive squarions other than 1.

The argument above show that there are none greater than 184, and a simple check shows that 1 is the only squarion below 184.

This result applies to any function $F$ defined for non-negative integers and taking non-negative integer values, including such weird functions as the following: $F(n)=n\lfloor n\sin(n)\rfloor$ :

Only $4 \textrm{ and } 5$ are F-ions for this choice of $F$ .

Thanks to Alexander Bogomolny for helpful discussions.

van Berkel, D. (2009) On a curious property of 3435. Retrieved from arxiv.org: On_a_curious_property_of_3435 [This article provides the argument I have described in this post]

Tagged as: factorions, Munchausen numbers

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Thursday, 2 June 2011

Vaktorions wekobions monchosin numbers and related matters

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