Palindromic Powers of Two

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.

  1. Start with a keyspace of 0 and all of the integers in order as your available keys.
  2. Write a 0.
  3. You’ve exhausted the keyspace, so get another key, 1.
  4. Write the entire series so far, add the new key, and write the series (minus the new key) again. You should have 010.
  5. You’ve exhausted the keyspace again, so get another key, 2.
  6. Repeat: 010, 2, 010 -> 0102010.
  7. Repeat with 3: 010201030102010.
  8. 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 20
2 21
3 20 * 3
4 22
5 20 * 5
6 21 * 3
7 20
8 23
9 20 * 32
10 21 * 5
11 20
12 22 * 3
13 20
14 21 * 7
15 20 * 3 * 5
16 24

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