{"app":{"locale":"en-GB"},"problem":{"content":"

A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.

For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$.
For example, $f_1(2)=2$, $f_1(4)=2$, $f_1(18)=6$ and $f_2(18)=36$.

Let $\\displaystyle S_k(n)=\\sum_{i=1}^{n} f_k(i)$.For example, $S_1(10)=41$, $S_1(100)=3512$, $S_2(100)=208090$, $S_1(10000)=35252550$ and $\\displaystyle \\sum_{k=1}^{3} S_k(10^{8}) \\equiv 338787512 \\pmod{ 1\\,000\\,000\\,007}$.

Find $\\displaystyle \\sum_{k=1}^{50} S_k(10^{12}) \\bmod 1\\,000\\,000\\,007$.

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