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Problem 773

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Ruff Numbers

Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$.

Define a $k$-Ruff number to be one that is not divisible by any element in $S_k$.

If $N_k$ is the product of the numbers in $S_k$ then define $F(k)$ to be the sum of all $k$-Ruff numbers less than $N_k$ that have last digit $7$. You are given $F(3) = 76101452$.

Find $F(97)$, give your answer modulo $1\ 000\ 000\ 007$.

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仿粗糙数

记$S_k$为$2$、$5$和前$k$个以$7$结尾的素数所构成的集合。例如，$S_3 = \{2,5,7,17,37\}$。

如果一个数不能被$S_k$中的任意元素整除，则称之为$k$-仿粗糙数。

若$N_k$为$S_k$中的元素之积，则定义$F(k)$为所有小于$N_k$且以$7$结尾的$k$-仿粗糙数之和。已知$F(3) = 76101452$。

求$F(97)$，并将你的答案对$1\ 000\ 000\ 007$取余。

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