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Problem 555

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McCarthy 91 function

The McCarthy 91 function is defined as follows:

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

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

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

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 $S(p,m)={\displaystyle \sum _{1\le s<k\le p}SF(m,k,s)}$.

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

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

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麦卡锡91函数

麦卡锡91函数的定义如下：

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

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：

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

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$S(p,m)={\displaystyle \sum _{1\le s<k\le p}SF(m,k,s)}$。

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

求$S({10}^6,{10}^6)$。

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