Factorions

The four numbers $latex 1, decease 2, treatment 145 extrm{ and } 40585$ have the following peculiar property:

$latex 1!=1, clinic 2!=2, 1!+4!+5!=145 extrm{ and } 4!+0!+5!+8!+5! =40585$.

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.

Münchausen numbers

The numbers have the peculiar property that .

Numbers (base 10 digits) with the property  that are called Münchausen numbers, and dependent on how we define – either as – there are 2 or 3 of them.

Cubions

Are there any positive integers (base 10 digits) with the property  that ?

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

F-ions

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

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

We call a non-negative integer (base 10 digits) an F-ion if 

Factorions correspond to the function , Münchausen numbers to the function (with an appropriate choice of ), and cubions to the function .

Notation: for (base 10 digits) we will denote by

Bounds on the number of F-ions

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

This allows us to conclude that for such 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.

1. The number of digits of a positive integer

For a positive integer the number of (base 10) digits of is the floor of , the largest integer less than or equal to .

This means that the number of digits of is less than .

2. An inequality for

If the base 10 digits of the positive integer are then:

1. for
2. The number of digits of is

Let

Then we have:

This says that cannot grow too big in terms of the number of digits of .

3. An inequality for large

We look at the behavior of the function :

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

The derivative of is and this is positive for so for the function is strictly increasing.

Therefore there is a smallest non-negative integer for which .

For we have and so .

So, for we cannot have , so showing that the number of F-ions is finite.

Finding a solution to

The least non-negative integer for which is the ceiling of the unique solution to .

There are a number of ways to estimate the solution to .

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 we first calculate and guesstimate from a graph of that :

The Mathematica code:

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

gives so .

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

Squarions

A squarion is a positive integer for which where .

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.

Further thoughts

This result applies to any function defined for non-negative integers and taking non-negative integer values, including such weird functions as the following:  :

Only are F-ions for this choice of .

Acknowledgements

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]