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Problem 704

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Factors of Two in Binomial Coefficients

Define $g(n, m)$ to be the largest integer $k$ such that $2^k$ divides $\binom{n}m$. For example, $\binom{12}5 = 792 = 2^3 \cdot 3^2 \cdot 11$, hence $g(12, 5) = 3$. Then define $F(n) = \max { g(n, m) : 0 \le m \le n }$. $F(10) = 3$ and $F(100) = 6$.

Let $S(N)$ = $\displaystyle\sum_{n=1}^N{F(n)}$. You are given that $S(100) = 389$ and $S(10^7) = 203222840$.

Find $S(10^{16})$.

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二项式系数中的质因数二

记$g(n, m)$为使得$2^k$整除$\binom{n}m$的最大整数$k$。例如，$\binom{12}5 = 792 = 2^3 \cdot 3^2 \cdot 11$，因此$g(12, 5) = 3$。再定义$F(n) = \max { g(n, m) : 0 \le m \le n }$。已知$F(10) = 3$，$F(100) = 6$。

记$S(N)$ = $\displaystyle\sum_{n=1}^N{F(n)}$。已知$S(100) = 389$，$S(10^7) = 203222840$。

求$S(10^{16})$。

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