Let $\sigma(n)$ be the sum of all the divisors of the positive integer $n$, for example:
$\sigma(10) = 1+2+5+10 = 18$.
Define $T(N)$ to be the sum of all numbers $n \le N$ such that when the fraction $\frac{\sigma(n)}{n}$ is written in its lowest form $\frac ab$, the denominator is a power of 3 i.e. $b = 3^k, k > 0$.
You are given $T(100) = 270$ and $T(10^6) = 26089287$.
Find $T(10^{14})$.
Solution
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