Republic of Mathematics blog

Factorions, Munchausen numbers, cubions and related things

Posted by: Gary Ernest Davis on: April 28, 2011

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 1 	extrm{ and } 3435 have the peculiar property that 1^1=1 	extrm{ and } 3^3+4^4+3^3+5^5.

Numbers n=d_kldots 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 	extrm{ or } 0 – there are 2 or 3 of them.

Cubions

Are there any positive integers n=d_kldots 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.

F-ions

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_kldots 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_kldots d_1d_0 (base 10 digits) we will denote F(d_k)+ldots F(d_0)=n by n^{[F]}

Bounds on the number of F-ions

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.

1. The number of digits of a positive integer

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.

2. An inequality for n^{[F]}

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

  1. 0leq d_i leq 9 for 0leq ileq k
  2. The number of digits of n is k+1 leq log_{10}(n)+1

Let A	extrm{ 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.

3. An inequality for large 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 ngeq 3 the function n	o 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 ngeq n^* we have frac{n}{log_{10}(n)+1}geq frac{n^*}{log_{10}(n^*)+1}>A and so ngeq A (log_{10}(n)+1).

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

Finding a solution to frac{x}{log_{10}(x)+1}=A

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 xapprox 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.

Squarions

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.

Further thoughts

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)=nlfloor nsin(n) floor :

Only 4 	extrm{ and } 5 are F-ions for this choice of F.

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]

2 Responses to "Factorions, Munchausen numbers, cubions and related things"

What happens if you replace base ten with something different: eg factorial representation? Presumably it might be possible then to find F for which there are infinitely many F-ions?

It should not come as a surprise that I liked your post.

I also though about generalizing the Munchausen numbers and came up with the same conclusions as that you demonstrated in this post.

It did get my thinking on the following question: what function would produce the most F-ions? In order to compare different function “honestly” one should normalize the count by dividing by the smallest upper bound.

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