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^{0} * 3
4 2^{2}
5 2^{0} * 5
6 2^{1} * 3
7 2^{0}
8 2^{3}
9 2^{0} * 3^{2}
10 2^{1} * 5
11 2^{0}
12 2^{2} * 3
13 2^{0}
14 2^{1} * 7
15 2^{0} * 3 * 5
16 2^{4}