Problem 1327

1. Define the functions and the problem statement:
- Let $\varphi(m)$ be the Euler's totient function, which counts the number of positive integers up to $m$ that are relatively prime to $m$.
- Let $\sigma(m)$ be the sum of the positive divisors of $m$.
- We need to find the sum of all even positive integers $n$ such that
$$\frac{n^5 \sigma(n) - 2}{\varphi(n)}$$
is an integer.

2. **Express $n$ in terms of its prime factors:**
- Let $n = 2^a p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_i$ are odd primes.
- The Euler's totient function for $n$ is:
$$\varphi(n) = \varphi(2^a) \varphi(p_1^{a_1}) \varphi(p_2^{a_2}) \cdots \varphi(p_k^{a_k})$$
Since $\varphi(2^a) = 2^a - 2^{a-1} = 2^{a-1}$ for $a \geq 1$, and $\varphi(p_i^{a_i}) = p_i^{a_i} - p_i^{a_i-1}$.

3. Analyze the divisibility condition:
- We need $\frac{n^5 \sigma(n) - 2}{\varphi(n)}$ to be an integer.
- Consider the 2-adic valuation $v_2$ of $\varphi(n)$:
$$v_2(\varphi(n)) = (a-1) + v_2(\varphi(p_1^{a_1})) + v_2(\varphi(p_2^{a_2})) + \cdots + v_2(\varphi(p_k^{a_k}))$$
Since $p_i$ are odd primes, $v_2(\varphi(p_i^{a_i})) = 0$. Thus,
$$v_2(\varphi(n)) = a - 1$$

4. **Evaluate $v_2(n^5 \sigma(n) - 2)$:**
- For $n = 2^a p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, $n^5 \sigma(n)$ is even, so $n^5 \sigma(n) - 2$ is odd.
- Therefore, $v_2(n^5 \sigma(n) - 2) = 1$.

5. Set up the inequality:
- We need $v_2(n^5 \sigma(n) - 2) \geq v_2(\varphi(n))$:
$$1 \geq a - 1$$
This implies:
$$a \leq 2$$

6. **Consider possible values of $a$ and $k$:**
- If $a = 1$, then $n = 2p$ where $p$ is an odd prime.
- If $a = 2$, then $n = 4$.

7. Evaluate specific cases:
- For $n = 2p$:
$$\varphi(2p) = p-1, \quad \sigma(2p) = (1+2)(1+p) = 3(p+1)$$
We need:
$$\frac{(2p)^5 \cdot 3(p+1) - 2}{p-1} \text{ to be an integer}$$
Simplifying:
$$\frac{96p^6 + 96p^5 - 2}{p-1}$$
This implies $p-1$ must divide 190. Possible values of $p-1$ are 2, 10, 190, giving $p = 3, 11, 191$.

- For $n = 4$:
$$\varphi(4) = 2, \quad \sigma(4) = 1 + 2 + 4 = 7$$
We need:
$$\frac{4^5 \cdot 7 - 2}{2} = \frac{7166}{2} = 3583 \text{ which is an integer}$$

8. **Sum the valid $n$ values:**
- Valid $n$ values are $2, 4, 6, 22, 382$.
- Sum: $2 + 4 + 6 + 22 + 382 = 416$.