---

Problem 555

---

McCarthy 91 function

The McCarthy 91 function is defined as follows:

$M_{91}(n) = \begin{cases} n - 10 & \text{if } n > 100 \\ M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100 \end{cases}$

We can generalize this definition by abstracting away the constants into new variables:

$M_{m,k,s}(n) = \begin{cases} n - s & \text{if } n > m \\ M_{m,k,s}(M_{m,k,s}(n+k)) & \text{if } 0 \leq n \leq m \end{cases}$

This way, we have $M_{91} = M_{100,11,10}$.

Let $F_{m,k,s}$ be the set of fixed points of $M_{m,k,s}$. That is,

$F_{m,k,s}= \{ n \in \mathbb{N} , | , M_{m,k,s}(n) = n \}$

For example, the only fixed point of $M_{91}$ is $n = 91$. In other words, $F_{100,11,10}= \{91\}$.

Now, define $SF(m,k,s)$ as the sum of the elements in $F_{m,k,s}$ and let $S(p,m) = \displaystyle \sum_{1 \leq s < k \leq p}{SF(m,k,s)}$.

For example, $S(10, 10) = 225$ and $S(1000, 1000)=208724467$.

Find $S(10^6, 10^6)$.

---

麦卡锡91函数

麦卡锡91函数的定义如下：

$M_{91}(n) = \begin{cases} n - 10 & \text{if } n > 100 \\ M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100 \end{cases}$

通过将上述定义中的常数抽象为变量，我们可以将这个定义一般化：

$M_{m,k,s}(n) = \begin{cases} n - s & \text{if } n > m \\ M_{m,k,s}(M_{m,k,s}(n+k)) & \text{if } 0 \leq n \leq m \end{cases}$

这样一来，我们有$M_{91} = M_{100,11,10}$。

令$F_{m,k,s}$为$M_{m,k,s}$的不动点所构成的集合，也就是说：

$F_{m,k,s}= \{ n \in \mathbb{N} , | , M_{m,k,s}(n) = n \}$

例如，$M_{91}$只有一个不动点$n = 91$，换言之就是$F_{100,11,10}= \{91\}$。

定义$SF(m,k,s)$为$F_{m,k,s}$中元素的和，并记$S(p,m) = \displaystyle \sum_{1 \leq s < k \leq p}{SF(m,k,s)}$。

已知$S(10, 10) = 225$以及$S(1000, 1000)=208724467$。

求$S(10^6, 10^6)$。

---