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Problem 637

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Flexible digit sum

Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$, and then carrying out the additions.

For example, from $n={123}_{10}$ ($n$ in base $10$) we can construct the four base $10$ integers ${123}_{10}$, $1+23={24}_{10}$, $12+3={15}_{10}$ and $1+2+3=6_{10}$.

Let $f(n,B)$ be the smallest number of steps needed to arrive at a single-digit number in base $B$. For example, $f(7,10)=0$ and $f(123,10)=1$.

Let $g(n,B_1,B_2)$ be the sum of the positive integers $i$ not exceeding $n$ such that $f(i,B_1)=f(i,B_2)$.

You are given $g(100,10,3)=3302$.

Find $g({10}^7,10,3)$.

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灵活的数字和

给定任意正整数$n$，我们可以通过在$n$的$B$进制表示中插入加号，并计算加法的结果，来构造新的整数。

例如，从$n={123}_{10}$ ($10$进制表示的$n$)出发，我们能够构造四个$10$进制整数：${123}_{10}$，$1+23={24}_{10}$，$12+3={15}_{10}$和$1+2+3=6_{10}$。

记$f(n,B)$为将$B$进制下的$n$变为一位数所需的最少步数。例如，$f(7,10)=0$而$f(123,10)=1$。

记$g(n,B_1,B_2)$为所有小于等于$n$且满足$f(i,B_1)=f(i,B_2)$的正整数$i$之和。

已知$g(100,10,3)=3302$。

求$g({10}^7,10,3)$。

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