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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