Sum of digits of sum of digits of powers of 12345 Sum of digits of $12345$ is $1+2+3+4+5=15$. The sum of digits of sum of digits is $1+5=6$.
I have plotted the sum of digits of powers of $12345$ with blue dots (x-axis is the power).
As the average digit is $4.5$, we can see the roughly linear rise with slope $log_{10}(12345) * 4.5 ≈ 18.4$
The red dots represent the sum of digits of sum of digits. Any hint why they are multiplies of $9$? (except the first red dot with value of $6$)


The sum of digits is divisible by 9 if and only if the number itself is divisible by 9. As 123456 is a multiple of 3 its square and all higher powers are divisible by 9. So the sum of digits, sum of sum of digits, etc. This is a standard test for divisibility by 9.
The reason this test works is as follows. Each digit $a_n$ of a number represents $10^n\times a_n$, where $n=0$ for the units, $n=1$ for the tens, etc. But each power $10^n=1+99...9$ ($(n$ nines). As $99...9$ is divisible by $9$ for any number of $9$s, the number itself is the sum of its digits plus a multiple of $9$. It follows that if the sum of digits is divisible by $9$, so is the number.
