Solution:

The relation says $5^{a_{n+1} - a_n} = 1 + \frac{3}{3n + 2} = \frac{3n+5}{3n+2}.$ Multiplying these equations for $n = 1, 2, \ldots, k - 1$ telescopes: $$5^{a_k - a_1} = \frac{3k + 2}{5}, \qquad \text{so} \qquad a_k = 1 + \log_5 \frac{3k+2}{5} = \log_5 (3k + 2).$$

Thus $a_k$ is an integer exactly when $3k + 2$ is a power of $5.$ Since $5^j \equiv 2^j \pmod 3,$ only odd exponents $j$ give numbers of the form $3k + 2.$ The power $5^1 = 5$ gives $k = 1,$ which is excluded, and the next, $5^3 = 125 = 3 \cdot 41 + 2,$ gives $k = 41.$