Problem 839

Beans in Bowls

The sequence $S_{n}$ is defined by $S_{0} = 290797$ and $S_{n} = S_{n - 1}^{2} mod 50515093$ for $n > 0$ .

There are $N$ bowls indexed $0, 1, \dots, N - 1$ . Initially there are $S_{n}$ beans in bowl $n$ .

At each step, the smallest index $n$ is found such that bowl $n$ has strictly more beans than bowl $n + 1$ . Then one bean is moved from bowl $n$ to bowl $n + 1$ .

Let $B (N)$ be the number of steps needed to sort the bowls into non-descending order.

For example, $B (5) = 0$ , $B (6) = 14263289$ and $B (100) = 3284417556$ .

Find $B (10^{7})$ .

碗中豆

序列 $S_{n}$ 按如下方式定义： $S_{0} = 290797$ ；对于 $n > 0$ ， $S_{n} = S_{n - 1}^{2} mod 50515093$ 。

有 $N$ 个碗，其编号分别为 $0, 1, \dots, N - 1$ 。一开始，编号为 $n$ 的碗中放有 $S_{n}$ 颗豆子。

接下来的每一步中，先选择编号最小的、豆子数目比后一个碗严格更多的碗 $n$ ，再从碗 $n$ 中移动一颗豆子到碗 $n + 1$ 。

记 $B (N)$ 为将碗中豆子数目调整为非递降序列所需的步数。

例如， $B (5) = 0$ ， $B (6) = 14263289$ ， $B (100) = 3284417556$ 。

求 $B (10^{7})$ 。