Maximizing Efficiency of Collatz Conjecture Program Python

Question

My question is very simple.

I wrote this program for pure entertainment. It takes a numerical input and finds the length of every Collatz Sequence up to and including that number.

I want to make it faster algorithmically or mathematically (i.e. I know I could make it faster by running multiple versions parallel or by writing it in C++, but where's the fun in that?).

Any and all help is welcome, thanks!

from matplotlib import pyplot as plt
from tqdm import tqdm
# Get Range to Check, Make Memory
top_range = int(input('Top Range: '))
mem = [0] * (top_range + 1)
def collatz():
 for start in tqdm(range(2, top_range + 1)):
 # If mod(4) == 1: Value 2 or 3 Cached
 if start % 4 == 1:
 mem[start] = mem[(start + (start >> 1) + 1) // 2] + 3
 # If mod(4) == 3: Use Algorithm
 elif start % 4 == 3:
 num = start
 count = 0
 while num >= start:
 if num % 2:
 num += (num >> 1) + 1
 count += 2
 else:
 num //= 2
 count += 1
 mem[start] = mem[num] + count
 # If mod(4) == 2 or 4: Value 1 Cached
 else:
 mem[start] = mem[(start // 2)] + 1
collatz()
# Plot each starting number with the length of it's sequence
plt.scatter([*range(1, len(mem) + 1)], mem, color = 'black', s = 1)
plt.show()