Properties of a special Digit/Sum function (1 Viewer)

[Sy123]

Hello all,

I decided to investigate a function that I thought of:



i.e. n=14

14 -> 9 -> 6 -> 5



Then I decided to iterate f(n), and as expected, if you keep doing it long enough, you will end up with 1,2,3,5 or 7.

It turns out 1 and 3 can only appear once with a beginning n.

So if you iterate f(n) for all numbers from 0 to 1000, you get ~290 5's that appear.

I decided to investigate the amount of times that '5' appears, it would appear to me that the ratio between the number of numbers checked, and number of times 5 appears, will be periodic.

Indeed, this turns out to be the case:

Basically, I was trying to find (k/n) where k is the number of times 5 appears when you iterate all numbers from 1 to n.
n is the number we try, so if we say n=10

Then: try k=1, iterate, we get 1
try k=2, iterate, we get 2
try k=3, iterate, we get 3
try k=4, iterate, we 4
try k=5, iterate, we get 5
try k=6; iterate, we get 5
try k=7, iterate, we get 7
try k=8, iterate, we get 5
try k= 9, iterate, we get 5
try k=10, iterate, we get 7
So 5 appears 4 times, out of 10 times, so I would record 4/10

Using this code (java):

http://pastebin.com/V9kdKuBs

I got the following data: (~9300 lines of a .txt file)

https://www.dropbox.com/s/0e3yjkau7ff31d1/nums.txt

I put this into excel, and I tried to get good graph, to graph these results, I got:

Unfortunately, I think Eclipse had stopped after line 9300 (even though I told the program to do it for 10000 times), so I cannot get a larger sample space

But it looks to me, to be a steady increase (after the initial expected weirdness)

Now, that graph is ONLY if it ends in 5

I also wanted to get a feeling of what other results we can get.

So if we iterate all numbers from 1 to 101

1 appears 1 time
2 appears 28 times
3 appears 1 times
5 appears 24 times
7 appears 24 times

And if we iterate from 1 to 1001 we get similar ratios.

I have many questions

1) Is it possible to prove mathematically, whether the ratio of the number of times 5 appears, is periodic but is eventually increasing? Perhaps to some limiting value of 1/3?

2) What properties can be derived from such an iterating function

 

[GoldyOrNugget]

SO PROUD OF MY BOY

but

Your file doesn't list all the numbers up to 10k because your program terminates before it's finished writing. Close your PrintWriter with writ.close() when you're done with it.

Your factorList function is broken for multiples of 4. You want the for-loop limit to be n/2+1. This is screwing up your results badly.

What do you mean by periodic but increasing? The two contradict each other don't they? What's your defn of periodic?

I experimented a bit but didn't get much further than you. Here: http://nbviewer.ipython.org/gist/DanGoldbach/875b07cea8cc7f7c88ae

I don't think you're gonna be able to do much mathematical analysis of the properties of this function. I can't find many resources on the SumOfPrimeFactors function or the DigitSum function. There are some basic properties of each on OEIS and Wolfram Mathworld.

If I have any insights I'll update you

 

[Sy123]

SO PROUD OF MY BOY

but

Your file doesn't list all the numbers up to 10k because your program terminates before it's finished writing. Close your PrintWriter with writ.close() when you're done with it.

Your factorList function is broken for multiples of 4. You want the for-loop limit to be n/2+1. This is screwing up your results badly.

What do you mean by periodic but increasing? The two contradict each other don't they? What's your defn of periodic?

I experimented a bit but didn't get much further than you. Here: http://nbviewer.ipython.org/gist/DanGoldbach/875b07cea8cc7f7c88ae

I don't think you're gonna be able to do much mathematical analysis of the properties of this function. I can't find many resources on the SumOfPrimeFactors function or the DigitSum function. There are some basic properties of each on OEIS and Wolfram Mathworld.

If I have any insights I'll update you

Oh wow nice!

Thank you for your insights and suggestions to amend it, I'll try to experiment with more simple functions