Pick any whole number greater than 1.
1. Write down all of its proper divisors (including 1 and the number itself)
2. Add the digits of these divisors.
3. Use this sum to repeat steps 1 and 2 until your number does not change.
Must the process terminate?
At what number(s) can the process terminate at?
What numbers <1000 take the most steps to terminate?
The number of divisors of n < 2 Sqrt(n);
The number of digits in each divisor <= 1 + Log_10 (n);
The maximum value of each digit is 9.
If f(n) performs one iteration on n, then
f(n) < 9 (1+Log(n) ) 2 Sqrt(n)
For n >= 10000, f(n) < 9 (1+Log(n) ) 2 Sqrt(n) < n; The exact boundary is about 7744
Since the sequence converges to 15 for all 1<n<10000 and is strictly decreasing for all values above 10000, we can conclude that it converges to 15 for all n >1..
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Posted by goFish
on 2006-01-11 12:23:52 |
Proof?
