# Euler 0014

### The Problem:

We are trying to find the longest Collatz sequence under 1000000.

### Considerations:

The collatz sequence follows 2 simple rules:  
\- if n is even, then n/2  
\- if n is odd, then n\*3 + 1

5 -> 16 -> 8 -> 4 -> 2 -> 1   
6 -> 5 -> 4 -> 3 -> 2 -> 1

### The Approach:

The approach I'll take to this is to set up a dictionary of lengths at key n. If the key doesn't have a value then it generates a collatz sequence until it returns 1 or a key that has a length value

<div class="md-output__section has-scroll" id="bkmrk-keep-iterating-throu"></div>### The Code:

<details id="bkmrk-target_under-%3D-1000-"><summary></summary>

```python
highest_starting =  1000000
lengths_dict = {1:1}
max_length = 1
length_key = 1

#the recursive function to generate collatz.
#Base case is satisfied by the lengths_dict above already having 1 initialized.
def update_lengths(number, lengths):
    #print(number)
    if lengths.get(number) == None:
        if number % 2 == 0:
            one_down = number/2
        else:
            one_down = number*3 + 1
        lengths = update_lengths(one_down, lengths)
        lengths[number] = lengths[one_down] + 1

    return lengths

for i in range(1,highest_starting + 1):
    if lengths_dict.get(i) == None:
        lengths_dict = update_lengths(i, lengths_dict)
    
    if lengths_dict.get(i) > max_length:
        length_key = i
        max_length = lengths_dict.get(i)

print(length_key, max_length)

```

</details>