Math? I Thought This Was Computer Science!

Let \(f(a,b)\) be the largest integer \(c\) such that \(a \equiv 0 \pmod{b^c}\). For example, \(f(22,3) = 0\), \(f(42,7) = 1\), \(f(72,6) = 2\), and \(f(243,3) = 5\).

Let \[G(n) = \displaystyle \sum_{i=1}^n \sum_{j=2}^\infty f(i,j).\]

For example, \(G(25) = 76\), \(G(1000) = 7004\), and \(G(100000) = 1166220\).

Compute the value of \(G(10^8)\).

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