Matching Digit Sums

Published on Saturday, 22nd June 2019, 04:00 pm; Solved by 149

Problem 676

Let $d(i,b)$ be the digit sum of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$. When using different bases, the respective digit sums most of the time deviate from each other, for example $d(9,4)=3 \ne d(9,2)$.

However, for some numbers $i$ there will be a match, like $d(17,4)=d(17,2)=2$. Let $M(n,b_1,b_2)$ be the sum of all natural numbers $i \le n$ for which $d(i,b_1)=d(i,b_2)$. For example, $M(10,8,2)=18$, $M(100,8,2)=292$ and $M(10^6,8,2)=19173952$.

Find $\displaystyle \sum_{k=3}^6 \sum_{l=1}^{k-2}M(10^{16},2^k,2^l)$, giving the last 16 digits as the answer.