Powers of Persistence

“The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.” — OEIS

Base 5

23 → 11 → 1 in b5 (c=3) (n=13 in b10)
233 → 33 → 14 → 4 in b5 (c=4) (n=68 in b10)
33334 → 2244 → 224 → 31 → 3 in b5 (c=5) (n=2344 in b10)
444444444444 → 13243332331 → 333124 → 1331 → 14 → 4 in b5 (c=6) (n=244140624 in b10)
3344444444444444444444 → 2244112144242244414 → 13243332331 → 333124 → 1331 → 14 → 4 in b5 (c=7) (n=1811981201171874 in b10)


Base 6

23 → 10 → 0 in b6 (c=3) (n=15 in b10)
35 → 23 → 10 → 0 in b6 (c=4) (n=23 in b10)
444 → 144 → 24 → 12 → 2 in b6 (c=5) (n=172 in b10)
24445 → 2544 → 424 → 52 → 14 → 4 in b6 (c=6) (n=3629 in b10)


Base 7

24 → 11 → 1 in b7 (c=3) (n=18 in b10)
36 → 24 → 11 → 1 in b7 (c=4) (n=27 in b10)
245 → 55 → 34 → 15 → 5 in b7 (c=5) (n=131 in b10)
4445 → 635 → 156 → 42 → 11 → 1 in b7 (c=6) (n=1601 in b10)
44556 → 6666 → 3531 → 63 → 24 → 11 → 1 in b7 (c=7) (n=11262 in b10)
5555555 → 443525 → 6666 → 3531 → 63 → 24 → 11 → 1 in b7 (c=8) (n=686285 in b10)
444555555555555666 → 465556434443526 → 115443241155 → 256641 → 4125 → 55 → 34 → 15 → 5 in b7 (c=9) (n=1086400325525346 in b10)


Base 8

24 → 10 → 0 in b8 (c=3) (n=20 in b10)
37 → 25 → 12 → 2 in b8 (c=4) (n=31 in b10)
256 → 74 → 34 → 14 → 4 in b8 (c=5) (n=174 in b10)
2777 → 1256 → 74 → 34 → 14 → 4 in b8 (c=6) (n=1535 in b10)
333555577 → 3116773 → 5126 → 74 → 34 → 14 → 4 in b8 (c=7) (n=57596799 in b10)


Base 9

25 → 11 → 1 in b9 (c=3) (n=23 in b10)
38 → 26 → 13 → 3 in b9 (c=4) (n=35 in b10)
57 → 38 → 26 → 13 → 3 in b9 (c=5) (n=52 in b10)
477 → 237 → 46 → 26 → 13 → 3 in b9 (c=6) (n=394 in b10)
45788 → 13255 → 176 → 46 → 26 → 13 → 3 in b9 (c=7) (n=30536 in b10)
2577777 → 275484 → 13255 → 176 → 46 → 26 → 13 → 3 in b9 (c=8) (n=1409794 in b10)


Base 10

25 → 10 → 0 (c=3)
39 → 27 → 14 → 4 (c=4)
77 → 49 → 36 → 18 → 8 (c=5)
679 → 378 → 168 → 48 → 32 → 6 (c=6)
6788 → 2688 → 768 → 336 → 54 → 20 → 0 (c=7)
68889 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (c=8)
2677889 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (c=9)
26888999 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (c=10)
3778888999 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (c=11)
277777788888899 → 4996238671872 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (c=12)


Base 11

26 → 11 → 1 in b11 (c=3) (n=28 in b10)
3A → 28 → 15 → 5 in b11 (c=4) (n=43 in b10)
69 → 4A → 37 → 1A → A in b11 (c=5) (n=75 in b10)
269 → 99 → 74 → 26 → 11 → 1 in b11 (c=6) (n=317 in b10)
3579 → 78A → 46A → 1A9 → 82 → 15 → 5 in b11 (c=7) (n=4684 in b10)
26778 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 in b11 (c=8) (n=38200 in b10)
47788A → 86277 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 in b11 (c=9) (n=757074 in b10)
67899AAA → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 in b11 (c=10) (n=130757439 in b10)
77777889999 → 2AA174996A → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 in b11 (c=11) (n=199718348047 in b10)


Base 12

26 → 10 → 0 in b12 (c=3) (n=30 in b10)
3A → 26 → 10 → 0 in b12 (c=4) (n=46 in b10)
6B → 56 → 26 → 10 → 0 in b12 (c=5) (n=83 in b10)
777 → 247 → 48 → 28 → 14 → 4 in b12 (c=6) (n=1099 in b10)
AAB → 778 → 288 → A8 → 68 → 40 → 0 in b12 (c=7) (n=1571 in b10)
3577777799 → 3BA55B53 → 557916 → 5576 → 736 → A6 → 50 → 0 in b12 (c=8) (n=17902874277 in b10)


Base 13

27 → 11 → 1 in b13 (c=3) (n=33 in b10)
3B → 27 → 11 → 1 in b13 (c=4) (n=50 in b10)
5A → 3B → 27 → 11 → 1 in b13 (c=5) (n=75 in b10)
9A → 6C → 57 → 29 → 15 → 5 in b13 (c=6) (n=127 in b10)
27A → AA → 79 → 4B → 35 → 12 → 2 in b13 (c=7) (n=439 in b10)
8AC → 58B → 27B → BB → 94 → 2A → 17 → 7 in b13 (c=8) (n=1494 in b10)
35AB → 99C → 59A → 288 → 9B → 78 → 44 → 13 → 3 in b13 (c=9) (n=7577 in b10)
9BBB → 55B6 → 99C → 59A → 288 → 9B → 78 → 44 → 13 → 3 in b13 (c=10) (n=21786 in b10)
2999BBC → 591795 → 65B5 → 99C → 59A → 288 → 9B → 78 → 44 → 13 → 3 in b13 (c=11) (n=13274091 in b10)
28CCCCCC → 9B89B93 → 591795 → 65B5 → 99C → 59A → 288 → 9B → 78 → 44 → 13 → 3 in b13 (c=12) (n=168938314 in b10)
377AAAABCCC → 2833B38BCB → B588A8A → 777995 → 4B2CA → 4A64 → 58B → 27B → BB → 94 → 2A → 17 → 7 in b13 (c=13) (n=494196864368 in b10)


Base 14

27 → 10 → 0 in b14 (c=3) (n=35 in b10)
3C → 28 → 12 → 2 in b14 (c=4) (n=54 in b10)
5B → 3D → 2B → 18 → 8 in b14 (c=5) (n=81 in b10)
99 → 5B → 3D → 2B → 18 → 8 in b14 (c=6) (n=135 in b10)
359 → 99 → 5B → 3D → 2B → 18 → 8 in b14 (c=7) (n=667 in b10)
CCC → 8B6 → 29A → CC → A4 → 2C → 1A → A in b14 (c=8) (n=2532 in b10)
359AB → 55AA → CA8 → 4C8 → 1D6 → 58 → 2C → 1A → A in b14 (c=9) (n=130883 in b10)
CDDDD → 8CC8C → 2C436 → 8B6 → 29A → CC → A4 → 2C → 1A → A in b14 (c=10) (n=499407 in b10)
3ABBDDDD → DAAAD54 → 63DAC8 → 5BC1A → 2596 → 2A8 → B6 → 4A → 2C → 1A → A in b14 (c=11) (n=397912927 in b10)
488AABCCCDDD → 39A59889584 → A89DBD84 → 598D14C → 5BC1A → 2596 → 2A8 → B6 → 4A → 2C → 1A → A in b14 (c=12) (n=18693488093783 in b10)


Base 15

28 → 11 → 1 in b15 (c=3) (n=38 in b10)
3D → 29 → 13 → 3 in b15 (c=4) (n=58 in b10)
5E → 4A → 2A → 15 → 5 in b15 (c=5) (n=89 in b10)
28C → CC → 99 → 56 → 20 → 0 in b15 (c=6) (n=582 in b10)
8AE → 4EA → 275 → 4A → 2A → 15 → 5 in b15 (c=7) (n=1964 in b10)
5BBB → 1E8A → 4EA → 275 → 4A → 2A → 15 → 5 in b15 (c=8) (n=19526 in b10)
BBBCC → 3BBC9 → B939 → BD3 → 1D9 → 7C → 59 → 30 → 0 in b15 (c=9) (n=596667 in b10)
2999BDE → 3C9CE6 → 66B7C → 9CC9 → 36C9 → 899 → 2D3 → 53 → 10 → 0 in b15 (c=10) (n=30104309 in b10)
39BBCCCCCD → 41CBD6D4C → 23C96E6 → 66B7C → 9CC9 → 36C9 → 899 → 2D3 → 53 → 10 → 0 in b15 (c=11) (n=140410607143 in b10)


Base 16

28 → 10 → 0 in b16 (c=3) (n=40 in b10)
3E → 2A → 14 → 4 in b16 (c=4) (n=62 in b10)
5F → 4B → 2C → 18 → 8 in b16 (c=5) (n=95 in b10)
BB → 79 → 3F → 2D → 1A → A in b16 (c=6) (n=187 in b10)
2AB → DC → 9C → 6C → 48 → 20 → 0 in b16 (c=7) (n=683 in b10)
3DDE → 1BBA → 4BA → 1B8 → 58 → 28 → 10 → 0 in b16 (c=8) (n=15838 in b10)
379BDD → 55C77 → 396C → 798 → 1F8 → 78 → 38 → 18 → 8 in b16 (c=9) (n=3644381 in b10)


Base 17

29 → 11 → 1 in b17 (c=3) (n=43 in b10)
3F → 2B → 15 → 5 in b17 (c=4) (n=66 in b10)
5G → 4C → 2E → 1B → B in b17 (c=5) (n=101 in b10)
9F → 7G → 6A → 39 → 1A → A in b17 (c=6) (n=168 in b10)
CE → 9F → 7G → 6A → 39 → 1A → A in b17 (c=7) (n=218 in b10)
3DD → 1CE → 9F → 7G → 6A → 39 → 1A → A in b17 (c=8) (n=1101 in b10)
9CF → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=9) (n=2820 in b10)
2AFF → F9C → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=10) (n=12986 in b10)
55DDF → CF4G → 25EB → 55A → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=11) (n=446163 in b10)
39DDGG → DGCG7 → 35F54 → F9C → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=12) (n=5079174 in b10)
DEGGGG → 86DCDC → DGCG7 → 35F54 → F9C → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=13) (n=19710955 in b10)
6BBBBBEEF → 6FBEB7G8 → 5B39ACE → 1CED8G → 35F54 → F9C → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=14) (n=46650378808 in b10)
2BDDDDDEEEEEF → 1FBBBB76B714 → 6FBEB7G8 → 5B39ACE → 1CED8G → 35F54 → F9C → 5A5 → EC → 9F → 7G → 6A → 39 → 1A → A in b17 (c=15) (n=1570081251102035 in b10)


Base 18

29 → 10 → 0 in b18 (c=3) (n=45 in b10)
3F → 29 → 10 → 0 in b18 (c=4) (n=69 in b10)
5E → 3G → 2C → 16 → 6 in b18 (c=5) (n=104 in b10)
8D → 5E → 3G → 2C → 16 → 6 in b18 (c=6) (n=157 in b10)
2BB → D8 → 5E → 3G → 2C → 16 → 6 in b18 (c=7) (n=857 in b10)
2CEG → GAC → 5GC → 2H6 → B6 → 3C → 20 → 0 in b18 (c=8) (n=15820 in b10)
AABF → 2EGC → GAC → 5GC → 2H6 → B6 → 3C → 20 → 0 in b18 (c=9) (n=61773 in b10)
8GGHH → 5B8DE → DD2G → GC8 → 4D6 → H6 → 5C → 36 → 10 → 0 in b18 (c=10) (n=938627 in b10)
AAAAAAH → 8HGH28 → 5B8DE → DD2G → GC8 → 4D6 → H6 → 5C → 36 → 10 → 0 in b18 (c=11) (n=360129437 in b10)


Base 19

2A → 11 → 1 in b19 (c=3) (n=48 in b10)
3G → 2A → 11 → 1 in b19 (c=4) (n=73 in b10)
5F → 3I → 2G → 1D → D in b19 (c=5) (n=110 in b10)
AB → 5F → 3I → 2G → 1D → D in b19 (c=6) (n=201 in b10)
DH → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=7) (n=264 in b10)
2BC → DH → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=8) (n=943 in b10)
7BG → 37G → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=9) (n=2752 in b10)
DII → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=10) (n=5053 in b10)
4AAH → IFH → CDB → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=11) (n=31253 in b10)
3BGII → 15HGF → 2I9D → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=12) (n=472548 in b10)
EEFHH → 69GBI → 15HGF → 2I9D → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=13) (n=1926275 in b10)
ADEFFH → 2F7HHE → 69GBI → 15HGF → 2I9D → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=14) (n=26556906 in b10)
4ADDDDEEF → 3E7919IH → 2HH7FE → 69GBI → 15HGF → 2I9D → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=15) (n=77518543969 in b10)
9999999BBFHHHI → 6B41DG4CB3BG → H27A5F3D → 2F7HHE → 69GBI → 15HGF → 2I9D → BCD → 4E6 → HD → BC → 6I → 5D → 38 → 15 → 5 in b19 (c=16) (n=399503342991325867 in b10)


Base 20

2A → 10 → 0 in b20 (c=3) (n=50 in b10)
3H → 2B → 12 → 2 in b20 (c=4) (n=77 in b10)
6D → 3I → 2E → 18 → 8 in b20 (c=5) (n=133 in b10)
7J → 6D → 3I → 2E → 18 → 8 in b20 (c=6) (n=159 in b10)
DI → BE → 7E → 4I → 3C → 1G → G in b20 (c=7) (n=278 in b10)
6DE → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=8) (n=2674 in b10)
CGG → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=9) (n=5136 in b10)
2BHI → GGC → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=10) (n=20758 in b10)
CDGG → 4JGG → 28CG → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=11) (n=101536 in b10)
2DEGJ → DGCG → 4JGG → 28CG → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=12) (n=429939 in b10)
77BBHJ → BJ7D7 → GCGD → 4JGG → 28CG → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=13) (n=23612759 in b10)
BBBCEEHHHHH → 8DCB4G21J4 → 21ED4J4 → DGCG → 4JGG → 28CG → 7DC → 2EC → GG → CG → 9C → 58 → 20 → 0 in b20 (c=14) (n=118569903663157 in b10)


Base 21

2B → 11 → 1 in b21 (c=3) (n=53 in b10)
3I → 2C → 13 → 3 in b21 (c=4) (n=81 in b10)
6H → 4I → 39 → 16 → 6 in b21 (c=5) (n=143 in b10)
AK → 9B → 4F → 2I → 1F → F in b21 (c=6) (n=230 in b10)
GH → CK → B9 → 4F → 2I → 1F → F in b21 (c=7) (n=353 in b10)
4GI → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=8) (n=2118 in b10)
GII → BFI → 6F9 → 1HC → 9F → 69 → 2C → 13 → 3 in b21 (c=9) (n=7452 in b10)
5FHJ → 2CJC → C8C → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=10) (n=53296 in b10)
2BGIJ → CKKC → 64CI → BFI → 6F9 → 1HC → 9F → 69 → 2C → 13 → 3 in b21 (c=11) (n=498286 in b10)
FHKKK → AA5HI → GAJF → 4J89 → C8C → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=12) (n=3083912 in b10)
3BDGHJK → AHKKA3 → AA5HI → GAJF → 4J89 → C8C → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=13) (n=304907819 in b10)
6BBHIJJJJ → G1BHJ4DF → AHKKA3 → AA5HI → GAJF → 4J89 → C8C → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=14) (n=247765672579 in b10)
3DDGGGGGGGIIJ → 284GJDKAD63I → 5D65FHGK3 → 5BIB3KC → 1J6DC9 → H5JF → 2CJC → C8C → 2CI → KC → B9 → 4F → 2I → 1F → F in b21 (c=15) (n=26851272398708896 in b10)

The Number of the Creased

Here’s an idea for a story à la M.R. James. A middle-aged scholar opens some mail one morning and finds nothing inside one envelope but a strip of paper with the numbers 216348597 written on it in sinister red ink. Someone has folded the strip several times so that there are creases between groups of numbers, like this: 216|348|5|97. Wondering what the significance of the creases is, the scholar hits on the step of adding the numbers created by them:


216 + 348 + 5 + 97 = 666

After that… Well, I haven’t written the story yet. But that beginning raises an obvious question. Is there any other way of getting a Number of the Creased from 216348597? That is, can you get 666, the Number of the Beast, by dividing 216348597 in another way? Yes, you can. In fact, there are six ways of creating 666 by dividing-and-summing 216348597:


666 = 2 + 1 + 634 + 8 + 5 + 9 + 7
666 = 2 + 163 + 485 + 9 + 7
666 = 216 + 348 + 5 + 97
666 = 21 + 63 + 485 + 97
666 = 21 + 6 + 34 + 8 + 597
666 = 2 + 16 + 3 + 48 + 597


216348597 is a permutation of 123456789, so does 123456789 yield a Number of the Creased? Yes. Two of them, in fact:


666 = 123 + 456 + 78 + 9
666 = 1 + 2 + 3 + 4 + 567 + 89


And 987654321 yields another:


666 = 9 + 87 + 6 + 543 + 21


And what about other permutations of 123456789? These are the successive records:

Using 123456789

666 = 123 + 456 + 78 + 9
666 = 1 + 2 + 3 + 4 + 567 + 89 (c=2)


Using 123564789

666 = 12 + 3 + 564 + 78 + 9
666 = 123 + 56 + 478 + 9
666 = 1 + 2 + 3 + 564 + 7 + 89 (c=3)


Using 125463978

666 = 1 + 2 + 5 + 4 + 639 + 7 + 8
666 = 12 + 546 + 3 + 97 + 8
666 = 1 + 254 + 6 + 397 + 8
666 = 1 + 2 + 546 + 39 + 78 (c=4)


Using 139462578

666 = 13 + 9 + 4 + 625 + 7 + 8
666 = 139 + 462 + 57 + 8
666 = 1 + 394 + 6 + 257 + 8
666 = 1 + 39 + 46 + 2 + 578
666 = 13 + 9 + 4 + 62 + 578 (c=5)


Using 216348597

666 = 2 + 1 + 634 + 8 + 5 + 9 + 7
666 = 2 + 163 + 485 + 9 + 7
666 = 216 + 348 + 5 + 97
666 = 21 + 63 + 485 + 97
666 = 21 + 6 + 34 + 8 + 597
666 = 2 + 16 + 3 + 48 + 597 (c=6)



216348597 is the best of the bestial. No other permutation of 123456789 yields as many as six Numbers of the Creased.