# Reverssum

Here’s a simple sequence. What’s the next number?

1, 2, 4, 8, 16, 68, 100, ?

The rule I’m using is this: Reverse the number, then add the sum of the digits. So 1 doubles till it becomes 16. Then 16 becomes 61 + 6 + 1 = 68. Then 68 becomes 86 + 8 + 6 = 100. Then 100 becomes 001 + 1 = 2. And the sequence falls into a loop.

Reversing the number means that small numbers can get big and big numbers can get small, but the second tendency is stronger for the first few seeds:

• 1 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 2 → 4 → 8 → 16 → 68 → 100 → 2
• 3 → 6 → 12 → 24 → 48 → 96 → 84 → 60 → 12
• 4 → 8 → 16 → 68 → 100 → 2 → 4
• 5 → 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 6 → 12 → 24 → 48 → 96 → 84 → 60 → 12
• 7 → 14 → 46 → 74 → 58 → 98 → 106 → 608 → 820 → 38 → 94 → 62 → 34 → 50 → 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 8 → 16 → 68 → 100 → 2 → 4 → 8
• 9 → 18 → 90 → 18
• 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2

An 11-seed is a little more interesting:

11 → 13 → 35 → 61 → 23 → 37 → 83 → 49 → 107 → 709 → 923 → 343 → 353 → 364 → 476 → 691 → 212 → 217 → 722 → 238 → 845 → 565 → 581 → 199 → 1010 → 103 → 305 → 511 → 122 → 226 → 632 → 247 → 755 → 574 → 491 → 208 → 812 → 229 → 935 → 556 → 671 → 190 → 101 → 103 (11 leads to an 18-loop from 103 at step 26; total steps = 44)

Now try some higher bases:

• 1 → 2 → 4 → 8 → 15 → 57 → 86 → 80 → 15 (base=11)
• 1 → 2 → 4 → 8 → 14 → 46 → 72 → 34 → 4A → B6 → 84 → 58 → 96 → 80 → 14 (base=12)
• 1 → 2 → 4 → 8 → 13 → 35 → 5B → C8 → A6 → 80 → 13 (base=13)
• 1 → 2 → 4 → 8 → 12 → 24 → 48 → 92 → 36 → 6C → DA → C8 → A4 → 5A → B6 → 80 → 12 (base=14)
• 1 → 2 → 4 → 8 → 11 → 13 → 35 → 5B → C6 → 80 → 11 (base=15)
• 1 → 2 → 4 → 8 → 10 → 2 (base=16)

Does the 1-seed always create a short sequence? No, it gets pretty long in base-19 and base-20:

• 1 → 2 → 4 → 8 →  → 1D → DF → 3 → 4 → 107 → 709 → 914 → 424 → 42E → E35 → 54 → 5C → C7D → D96 → 6B3 → 3C7 → 7D6 → 6EE → E2 → 28 → 90B → B1A → A2E → E3 → 5A → A7B → B90 → AC→ DD → F1 → 2C → C → 2 → 40 → 8 (base=19)
• 1 → 2 → 4 → 8 →  → 1C → CE → F → 108 → 80A → A16 → 627 → 731 → 13 → 43 → 363 → 36F → F77 → 794 → 4A7 → 7B5 → 5CA → ADC → CF5 → 54 → 4B → B → 1 → 3F → F5E → E79 → 994 → 4AB → BB9 → 9D2 → 2ED → DFB → BC → CB → C1E → E2 → 49 → 96B → B7F → F94 → 4B3 → 3C2 → 2D0 → D → 3 → 51 → 1B → BD → EF → 3 → 4 → 5 → 71 → 1F → F → 7 → 95 → 63 → 3F → 1 → 2D → D (base=20)

Then it settles down again:

• 1 → 2 → 4 → 8 →  → 1B → BD → EE → 0 → 1B (base=21)
• 1 → 2 → 4 → 8 →  → 1A → AC → DA → BE → FE → 0 → 1A (base=22)
• 1 → 2 → 4 → 8 →  → 19 → 9B → C6 → 77 → 7 → C → EA → BF → E → 0 → 19 (base=23)

Base-33 is also short:

1 → 2 → 4 → 8 →  →  → 1 → 0 → 1 (base=33)

And so is base-35:

1 → 2 → 4 → 8 →  →  → 1 →  →  → F →  →  →  → 0 → 1 (base=35)

So what about base-34?

1 → 2 → 4 → 8 →  →  → 1 →  → 10 → 0 → 26 → 63 → 47 → 75 → 6E → E8A → A9C → CA7 → 7B7 → 7B → C → E →  →  →  → D → D → 9 → 9 →  → 1 → 21 → 23 → 341 → 14B → B4 → 59 → 96E → E74 → 485 → 58 → 95 → 5A → B8 → 8C → D → F →  →  → 9 → 92 → 29 → 93 → 3C → CA → A → 7 → 7 → 01 → 11F → F1 → 2F → F3 → 4 → 5 → 6 → 8 → A → C → E → F →  →  →  → 5 → 51 → 13 → 3 →  →  → 0C → C1 → 2D → D3 → 4 → 66 → 67 → 83 → 39D → D9 → A → C → E →  → 1 → 1A → A → 6 → 6 →  → 8 → 8 → 8 → 91 → 2 → 4C → C5E → E69 → 979 → 980 → 8 → 8 → 9 → C → E2 → 2 → 0 → 1 →  →  → E → F →  →  →  → 6 → 7 → 4 → 5 →  → 2 → 3 → 4 → 62 → 2E → E → C → D → 8 → 98 → 8 (1 leads to a 30-loop from 8 / 298 in base-34 at step 111; total steps = 141)

An alternative rule is to add the digit-sum first and then reverse the result. Now 8 becomes 8 + 8 = 16 and 16 becomes 61. Then 61 becomes 61 + 6 + 1 = 68 and 68 becomes 86. Then 86 becomes 86 + 8 + 6 = 100 and 100 becomes 001 = 1:

• 1 → 2 → 4 → 8 → 61 → 86 → 1
• 2 → 4 → 8 → 61 → 86 → 1 → 2
• 3 → 6 → 21 → 42 → 84 → 69 → 48 → 6
• 4 → 8 → 61 → 86 → 1 → 2 → 4
• 5 → 1 → 2 → 4 → 8 → 62 → 7 → 48 → 6 → 27 → 63 → 27
• 6 → 21 → 42 → 84 → 69 → 48 → 6
• 7 → 41 → 64 → 47 → 85 → 89 → 601 → 806 → 28 → 83 → 49 → 26 → 43 → 5 → 6 → 27 → 63 → 27
• 8 → 61 → 86 → 1 → 2 → 4 → 8
• 9 → 81 → 9
• 10 → 11 → 31 → 53 → 16 → 32 → 73 → 38 → 94 → 701 → 907 → 329 → 343 → 353 → 463 → 674 → 196 → 212 → 712 → 227 → 832 → 548 → 565 → 185 → 991 → 101 → 301 → 503 → 115 → 221 → 622 → 236 → 742 → 557 → 475 → 194→ 802 → 218 → 922 → 539 → 655 → 176 → 91 → 102 → 501 → 705 → 717 → 237 → 942 → 759 → 87 → 208 → 812 → 328 → 143 → 151 → 851 → 568 → 785 → 508 → 125 → 331 → 833 → 748 → 767 → 787 → 908 → 529 → 545 → 955 → 479 → 994 → 6102 → 1116 → 5211 → 225 → 432 → 144 → 351 → 63 → 27 → 63