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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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