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$S(10^6, 10^6)$。
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