# Sigma, Sigh

Consider the following sequence $$2\times3 , 2\times3\times5 , 2\times3\times5\times7 , ...$$.

In general let us define $$\omega(n)$$ as $\large{\omega(n) = \prod_{i=1}^{n} p_i}$ where $$p_i$$ is the $$i$$th prime number. How many digits are there in $\large{ \sigma_2( \omega( 1000 ) ) }$

Details and assumptions

• $$\sigma_2(n)$$ is the divisor function of the second order. It returns the sum of the square of the divisors of $$n$$.

• For example $$\sigma_2(10) = 1^2 + 2^2 + 5^2 + 10^2 = 130$$

• As an explicit example $$\sigma_2(\omega(5)) =7930000$$.

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