---

Problem 699

---

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 $T({10}^{14})$.

---

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$T({10}^{14})$。

---