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Problem 850

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Fractions of Powers

Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x\rfloor +\{x\}$, where $\lfloor x\rfloor$ (the floor function) is an integer, and $0\le \{x\}<1$.

For positive integers $k$ and $n$, define the function
$$f_k(n)=\sum _{i=1}^n\left\{\frac{i^k}{n}\right\}$$
For example, $f_5(10)=4.5$ and $f_7(1234)=616.5$.

Let
$$S(N)=\sum _{\begin{array}{c}k=1\\ k\text{ odd}\end{array}}^N\sum _{n=1}^N{f}_k(n)$$
You are given that $S(10)=100.5$ and $S({10}^3)=123687804$.

Find $\lfloor S(33557799775533)\rfloor$. Give your answer modulo $977676779$.

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幂的小数部分

任意正实数$x$都可以分解成整数部分和小数部分$\lfloor x\rfloor +\{x\}$，其中$\lfloor x\rfloor$（下取整函数）是一个整数，而$0\le \{x\}<1$。

对于正整数$k$和$n$，定义函数
$$f_k(n)=\sum _{i=1}^n\left\{\frac{i^k}{n}\right\}$$
例如，$f_5(10)=4.5$，$f_7(1234)=616.5$。

记
$$S(N)=\sum _{\begin{array}{c}k=1\\ k\text{为奇数}\end{array}}^N\sum _{n=1}^N{f}_k(n)$$
已知$S(10)=100.5$，$S({10}^3)=123687804$。

求$\lfloor S(33557799775533)\rfloor$，并将你的答案对$977676779$取余。

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