Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$?
To solve the problem, we need to find the smallest \( N \) such that \( a_N > 2017 \). We start by analyzing the sequence defined by \( a_1 = 1 \) and \( a_{n+1} = a_n + \lfloor \sqrt{a_n} \rfloor \).

1. **Initial Terms Calculation:**
   - \( a_1 = 1 \)
   - \( a_2 = a_1 + \lfloor \sqrt{a_1} \rfloor = 1 + \lfloor 1 \rfloor = 1 + 1 = 2 \)
   - \( a_3 = a_2 + \lfloor \sqrt{a_2} \rfloor = 2 + \lfloor \sqrt{2} \rfloor = 2 + 1 = 3 \)
   - \( a_4 = a_3 + \lfloor \sqrt{a_3} \rfloor = 3 + \lfloor \sqrt{3} \rfloor = 3 + 1 = 4 \)
   - \( a_5 = a_4 + \lfloor \sqrt{a_4} \rfloor = 4 + \lfloor \sqrt{4} \rfloor = 4 + 2 = 6 \)
   - \( a_6 = a_5 + \lfloor \sqrt{a_5} \rfloor = 6 + \lfloor \sqrt{6} \rfloor = 6 + 2 = 8 \)
   - \( a_7 = a_6 + \lfloor \sqrt{a_6} \rfloor = 8 + \lfloor \sqrt{8} \rfloor = 8 + 2 = 10 \)
   - \( a_8 = a_7 + \lfloor \sqrt{a_7} \rfloor = 10 + \lfloor \sqrt{10} \rfloor = 10 + 3 = 13 \)
   - \( a_9 = a_8 + \lfloor \sqrt{a_8} \rfloor = 13 + \lfloor \sqrt{13} \rfloor = 13 + 3 = 16 \)
   - \( a_{10} = a_9 + \lfloor \sqrt{a_9} \rfloor = 16 + \lfloor \sqrt{16} \rfloor = 16 + 4 = 20 \)

2. **General Pattern Observation:**
   We observe that the sequence grows approximately as \( a_n \approx n^2 \). To find the smallest \( N \) such that \( a_N > 2017 \), we need to estimate \( N \) such that \( N^2 > 2017 \).

3. **Estimate \( N \):**
   - \( \sqrt{2017} \approx 44.9 \)
   - Therefore, \( N \) should be at least 45.

4. **Verification:**
   We need to verify if \( a_{45} > 2017 \):
   - \( a_{45} \approx 45^2 = 2025 \)
   - Since \( 2025 > 2017 \), we need to check if \( a_{44} \leq 2017 \).

5. **Check \( a_{44} \):**
   - \( a_{44} \approx 44^2 = 1936 \)
   - Since \( 1936 < 2017 \), we confirm that \( a_{45} \) is indeed the first term greater than 2017.

The final answer is \( \boxed{45} \).