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Problem 708

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Twos are all you need

A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we define $f(1)=1$.

For example, $90 = 2\times 3\times 3\times 5$, then replacing the primes, $2\times 2\times 2\times 2 = 16$, hence $f(90) = 16$.

Let $\displaystyle S(N)=\sum_{n=1}^{N} f(n)$. You are given $S(10^8)=9613563919$.

Find $S(10^{14})$.

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只需要二

对正整数$n$作质因数分解，并记$f(n)$为将所有质因数均替换为$2$时的新乘积；此外记$f(1)=1$。

例如，$90 = 2\times 3\times 3\times 5$，将质因数替换后得$2\times 2\times 2\times 2 = 16$，因此$f(90) = 16$。

记$\displaystyle S(N)=\sum_{n=1}^{N} f(n)$。已知$S(10^8)=9613563919$。

求$S(10^{14})$。

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