Huh, that’s odd.

Just farting around, reducing the set of Integers to their prime factors—typically Friday-night stuff, amirite?—and I noticed an interesting pattern.

- Start with a keyspace of 0 and all of the integers in order as your available keys.
- Write a 0.
- You’ve exhausted the keyspace, so get another key, 1.
- Write the entire series so far, add the new key, and write the series (minus the new key) again. You should have 010.
- You’ve exhausted the keyspace again, so get another key, 2.
- Repeat: 010, 2, 010 -> 0102010.
- Repeat with 3: 010201030102010.
- Repeat with 4: 0102010301020104010201030102010.

And so on.

The “Huh!” moment is this: you’re writing the exponent of two in the prime factorization of the list of integers:

1 **2 ^{0}**

2

**2**

^{1}3

**2*** 3

^{0}4

**2**

^{2}5

**2*** 5

^{0}6

**2*** 3

^{1}7

**2**

^{0}8

**2**

^{3}9

**2*** 3

^{0}^{2}

10

**2*** 5

^{1}11

**2**

^{0}12

**2*** 3

^{2}13

**2**

^{0}14

**2*** 7

^{1}15

**2*** 3 * 5

^{0}16

**2**

^{4}