What is the sum of sum of digits of $4444^{4444^{4444}}$?

Solution 1:

You can find an upper bound for it even without using computer or any calculator: $$ f(N) < 9 (4444^{4444} \times \log_{10} 4444 + 1) < 9 \times 4 \times 4444^{4444} + 9 $$ $$ f(f(N)) < 9 ( \log_{10}9 + \log_{10}4 + 4444 \log_{10}4444 + 1) < 9 (3 + 4444 \times 4) = 9 \times 17779 = 160011 $$

so $$ f(f(N))<160011 $$ this is a large range but it can be smaller with calculator. (Note that you should have computed base 10 logarithm instead of natural logarithm)

the range consists of 160011 numbers, and by knowing the reminder of 9, only 17,779 numbers are left, and answer is one of them.

Of course this is not an exact answer, but it is straightforward!

Edit: I just misused a formula which is about something different, sorry for that!