Problem 175

Fractions involving the number of different ways a number can be expressed as a sum of powers of 2

Define $f (0) = 1$ and $f (n)$ to be the number of ways to write $n$ as a sum of powers of $2$ where no power occurs more than twice.

For example, $f (10) = 5$ since there are five different ways to express $10$ :
$10 = 8 + 2 = 8 + 1 + 1 = 4 + 4 + 2 = 4 + 2 + 2 + 1 + 1 = 4 + 4 + 1 + 1$

It can be shown that for every fraction $p / q (p > 0, q > 0)$ there exists at least one integer $n$ such that $f (n) / f (n - 1) = p / q$ .

For instance, the smallest $n$ for which $f (n) / f (n - 1) = 13 / 17$ is $241$ .

The binary expansion of $241$ is $11110001$ .

Reading this binary number from the most significant bit to the least significant bit there are $4$ one’s, $3$ zeroes and $1$ one. We shall call the string $4, 3, 1$ the Shortened Binary Expansion of $241$ .

Find the Shortened Binary Expansion of the smallest $n$ for which
$f (n) / f (n - 1) = 123456789 / 987654321$ .

Give your answer as comma separated integers, without any whitespaces.

与幂和表示有关的分数

记 $f (0) = 1$ ， $f (n)$ 为将 $n$ 写成 $2$ 的幂次的和且任意幂次出现不超过两次的方式数。

例如， $f (10) = 5$ ，因为 $10$ 恰好有 $5$ 种不同的表示方式：
$10 = 8 + 2 = 8 + 1 + 1 = 4 + 4 + 2 = 4 + 2 + 2 + 1 + 1 = 4 + 4 + 1 + 1$

对于任意分数 $p / q$ （其中整数 $p > 0$ ，整数 $q > 0$ ），我们都能找到至少一个整数 $n$ ，使得 $f (n) / f (n - 1) = p / q$ 。

例如，使得 $f (n) / f (n - 1) = 13 / 17$ 的最小的 $n$ 是 $241$ 。

$241$ 的二进制表示为 $11110001$ 。

从左往右读这个二进制串我们得到 $4$ 个 $1$ ， $3$ 个 $0$ ，再 $1$ 个 $1$ 。因此，我们称数字串 $4, 3, 1$ 是 $241$ 的简式二进制表示。

找出满足下式的最小的 $n$ 的简式二进制表示：
$f (n) / f (n - 1) = 123456789 / 987654321$ .

你的答案应当用半角逗号“,”隔开各个整数，且没有任何空格。