ROSECODE 513

Runs Resistance

Min_25 · Math · 2018-09-29

Let $A_{0} (n)$ be the array of the binary representation of positive integer $n$ .

For example, $A_{0} (1) = [1]$ , $A_{0} (2) = [1, 0]$ and $A_{0} (11) = [1, 0, 1, 1]$ .

Let $R (A)$ be the function that returns the array of run-lengths of the given array $A$ .

For example, $R ([1, 0]) = [1, 1]$ , $R ([1, 0, 1, 1]) = [1, 1, 2]$ and $R ([1, 0, 0, 0, 1]) = [1, 3, 1]$ .

Let $A_{k} (n) := R (A_{k - 1} (n))$ for $k \geq 1$ , and $M (n)$ be the minimum integer $k$ ( $\geq 0$ ) such that $| A_{k} (n) | = 1$ , where $| A |$ denotes the length of $A$ .

Finally, let $S (n) := \sum_{i = 1}^{n} M (i)$ .

You can verify that $M (1) = 0$ , $M (2) = 2$ , $M (3) = 1$ , $M (11) = 4$ , $S (100) = 346$ and $S (1234567) = 6717538$ .

Find $S (1234567654321)$ .