I have implemented Shor's algorithm for $N=15$ from this tutorial. I understand the algorithm pretty well, but I'm a little confused at the output I'm getting from running the circuit. It appears to be all over the place. The function $\text{qpe_amod15}$ seems to take the zero-th reading and use that phase and extracted period, which has resulted in successful factorizations so far.
Does this look normal or am I just getting lucky?
]
This looks quite odd, so I would say that something is wrong with your implementation and that you are just getting lucky. (Note that for such a small example, it is quite probable that you would get lucky, even if the quantum circuit only outputs random values.)
You would expect to observe a frequency $j \in \{ 0, 64, 128, 192 \}$, if your control register is of length $8$ qubits, and your generator is $g = 7$, which has order $r = 4$ in $\mathbb Z_N^*$ where $N = 15$.
This answer to a similar question may be useful to build up your understanding. When the control register is of length $8$ qubits instead of $6$ qubits as in said answer, the frequencies are scaled up by a factor of $2^{8-6} = 4$, but otherwise the situation is analogous.
