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Problem 699

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Triffle Numbers

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 $\frac{a}{b}$, 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 $T(10^{14})$.

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三友数

记$\sigma(n)$为正整数$n$的所有因数之和，例如：$\sigma(10) = 1+2+5+10 = 18$。

考虑所有正整数$n \le N$，并将$\frac{\sigma(n)}{n}$写成既约分数$\frac{a}{b}$，记所有使得分母$b$为$3$的幂的数$n$之和为$T(N)$。

已知$T(100) = 270$，$T(10^6) = 26089287$。

求$T(10^{14})$。

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