Problem 988

Non-attacking Frogs

Frogs can be placed on the real number line at integer locations. Given coprime positive integers $(a,b)$, each frog has the ability to make jumps of distances $a$ or $b$ in the positive direction.

Two frogs placed at $m$ and $n$, $m<n$, are attacking if the frog at $m$ can hop to $n$ with some series of jumps. For example if $(a,b)=(3,5)$, frogs placed at $0$ and $11$ are attacking as the former can make two jumps of $3$ and one jump of $5$ to reach $11$. However, frogs placed at $4$ and $11$ are non-attacking.

A non-attacking configuration is a placement of any number of frogs such that:

one frog is placed at $0$;
all other frogs are placed at distinct positive integers;
no two frogs are attacking.

Define $F(a,b)$ to be sum of the integer locations of every frog, summing over all non-attacking configurations. For example if $(a,b)=(3,5)$ there are seven non-attacking configurations:
$$\{0\}\quad\quad\{0,1\}\quad\quad\{0,2\}\quad\quad\{0,4\}\quad\quad\{0,7\}\quad\quad\{0,1,2\}\quad\quad\{0,2,4\}
$$giving $F(3,5)=23$.

You are also given $F(5,13)=16336$.

Find $F(19,53)$.

蛙不见蛙

一些青蛙被放置在实数轴的整数位置上；给定一对互质的正整数$(a,b)$，每只青蛙每次能沿正方向跳出$a$或$b$的距离。

对于放置在$m$和$n$（$m<n$）处的两只青蛙，如果位于$m$的青蛙可以通过一系列跳跃到达$n$，则称它们是敌对的。例如，若$(a,b)=(3,5)$，则放置在$0$和$11$处的青蛙是敌对的，因为前者可以先跳两次距离$3$再跳一次距离$5$到达$11$；而放置在$4$和$11$处的青蛙则不是敌对的。

若一种放置任意只青蛙的方案满足以下条件，则称之为蛙不见蛙：

一只青蛙放置在$0$处；
所有其它青蛙放置在不同的正整数位置上；
任意两只青蛙均不敌对。

定义$F(a,b)$为所有蛙不见蛙方案中每只青蛙所在整数位置的总和。例如，若$(a,b)=(3,5)$，共有七种蛙不见蛙方案：
$$\{0\}\quad\quad\{0,1\}\quad\quad\{0,2\}\quad\quad\{0,4\}\quad\quad\{0,7\}\quad\quad\{0,1,2\}\quad\quad\{0,2,4\}$$
由此可得$F(3,5)=23$。

已知$F(5,13)=16336$。

求$F(19,53)$。