Rollercoaster Rules

n += digsum(n). It’s one of my favorite integer sequences — a rollercoaster to infinity. It works like this: you take a number, sum its digits, add the sum to the original number, and repeat:


1 → 2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77 → 91 → 101 → 103 → 107 → 115 → 122 → 127 → 137 → 148 → 161 → 169 → 185 → 199 → 218 → 229 → 242 → 250 → 257 → 271 → 281 → 292 → 305 → 313 → 320 → 325 → 335 → 346 → 359 → 376 → 392 → 406 → 416 → 427 → 440 → 448 → 464 → 478 → 497 → 517 → 530 → 538 → 554 → 568 → 587 → 607 → 620 → 628 → 644 → 658 → 677 → 697 → 719 → 736 → 752 → 766 → 785 → 805 → 818 → 835 → 851 → 865 → 884 → 904 → 917 → 934 → 950 → 964 → 983 → 1003 → 1007 → 1015 → 1022 → 1027 → 1037 → 1048 → 1061 → 1069 → 1085 → 1099 → 1118 → 1129 → 1142 → 1150 → 1157 → 1171 → 1181 → 1192 → 1205 → ...

I call it a rollercoaster to infinity because the digit-sum constantly rises and falls as n gets bigger and bigger. The most dramatic falls are when n gets one digit longer (except on the first occasion):


... → 8 (digit-sum=8) → 16 (digit-sum=7) → ...
... → 91 (ds=10) → 101 (ds=2) → ...
... → 983 (ds=20) → 1003 (ds=4) → ...
... → 9968 (ds=32) → 10000 (ds=1) → ...
... → 99973 (ds=37) → 100010 (ds=2) → ...
... → 999959 (ds=50) → 1000009 (ds=10) → ...
... → 9999953 (ds=53) → 10000006 (ds=7) → ...
... → 99999976 (ds=67) → 100000043 (ds=8) → ...
... → 999999980 (ds=71) → 1000000051 (ds=7) → ...
... → 9999999962 (ds=80) → 10000000042 (ds=7) → ...
... → 99999999968 (ds=95) → 100000000063 (ds=10) → ...
... → 999999999992 (ds=101) → 1000000000093 (ds=13) → ...

Look at 9968 → 10000, when the digit-sum goes from 32 to 1. That’s only the second time that digsum(n) = 1 in the sequence. Does it happen again? I don’t know.

And here’s something else I don’t know. Suppose you introduce a rule for the rollercoaster of n += digsum(n). You buy a ticket with a number on it: 1, 2, 3, 4, 5… Then you get on the rollercoaster powered by with that number. Now here’s the rule: Your ride on the rollercoaster ends when n += digsum(n) yields a rep-digit, i.e., a number whose digits are all the same. Here are the first few rides on the rollercoaster:


1 → 2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
2 → 4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
3 → 6 → 12 → 15 → 21 → 24 → 30 → 33
4 → 8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
5 → 10 → 11
6 → 12 → 15 → 21 → 24 → 30 → 33
7 → 14 → 19 → 29 → 40 → 44
8 → 16 → 23 → 28 → 38 → 49 → 62 → 70 → 77
9 → 18 → 27 → 36 → 45 → 54 → 63 → 72 → 81 → 90 → 99
10 → 11
11 → 13 → 17 → 25 → 32 → 37 → 47 → 58 → 71 → 79 → 95 → 109 → 119 → 130 → 134 → 142 → 149 → 163 → 173 → 184 → 197 → 214 → 221 → 226 → 236 → 247 → 260 → 268 → 284 → 298 → 317 → 328 → 341 → 349 → 365 → 379 → 398 → 418 → 431 → 439 → 455 → 469 → 488 → 508 → 521 → 529 → 545 → 559 → 578 → 598 → 620 → 628 → 644 → 658 → 677 → 697 → 719 → 736 → 752 → 766 → 785 → 805 → 818 → 835 → 851 → 865 → 884 → 904 → 917 → 934 → 950 → 964 → 983 → 1003 → 1007 → 1015 → 1022 → 1027 → 1037 → 1048 → 1061 → 1069 → 1085 → 1099 → 1118 → 1129 → 1142 → 1150 → 1157 → 1171 → 1181 → 1192 → 1205 → 1213 → 1220 → 1225 → 1235 → 1246 → 1259 → 1276 → 1292 → 1306 → 1316 → 1327 → 1340 → 1348 → 1364 → 1378 → 1397 → 1417 → 1430 → 1438 → 1454 → 1468 → 1487 → 1507 → 1520 → 1528 → 1544 → 1558 → 1577 → 1597 → 1619 → 1636 → 1652 → 1666 → 1685 → 1705 → 1718 → 1735 → 1751 → 1765 → 1784 → 1804 → 1817 → 1834 → 1850 → 1864 → 1883 → 1903 → 1916 → 1933 → 1949 → 1972 → 1991 → 2011 → 2015 → 2023 → 2030 → 2035 → 2045 → 2056 → 2069 → 2086 → 2102 → 2107 → 2117 → 2128 → 2141 → 2149 → 2165 → 2179 → 2198 → 2218 → 2231 → 2239 → 2255 → 2269 → 2288 → 2308 → 2321 → 2329 → 2345 → 2359 → 2378 → 2398 → 2420 → 2428 → 2444 → 2458 → 2477 → 2497 → 2519 → 2536 → 2552 → 2566 → 2585 → 2605 → 2618 → 2635 → 2651 → 2665 → 2684 → 2704 → 2717 → 2734 → 2750 → 2764 → 2783 → 2803 → 2816 → 2833 → 2849 → 2872 → 2891 → 2911 → 2924 → 2941 → 2957 → 2980 → 2999 → 3028 → 3041 → 3049 → 3065 → 3079 → 3098 → 3118 → 3131 → 3139 → 3155 → 3169 → 3188 → 3208 → 3221 → 3229 → 3245 → 3259 → 3278 → 3298 → 3320 → 3328 → 3344 → 3358 → 3377 → 3397 → 3419 → 3436 → 3452 → 3466 → 3485 → 3505 → 3518 → 3535 → 3551 → 3565 → 3584 → 3604 → 3617 → 3634 → 3650 → 3664 → 3683 → 3703 → 3716 → 3733 → 3749 → 3772 → 3791 → 3811 → 3824 → 3841 → 3857 → 3880 → 3899 → 3928 → 3950 → 3967 → 3992 → 4015 → 4025 → 4036 → 4049 → 4066 → 4082 → 4096 → 4115 → 4126 → 4139 → 4156 → 4172 → 4186 → 4205 → 4216 → 4229 → 4246 → 4262 → 4276 → 4295 → 4315 → 4328 → 4345 → 4361 → 4375 → 4394 → 4414 → 4427 → 4444

The 11-ticket is much better value than the tickets for 1..10. Bigger numbers behave like this:


1252 → 4444
1253 → 4444
1254 → 888888
1255 → 4444
1256 → 4444
1257 → 888888
1258 → 4444
1259 → 4444
1260 → 9999
1261 → 4444
1262 → 4444
1263 → 888888
1264 → 4444
1265 → 4444
1266 → 888888
1267 → 4444
1268 → 4444
1269 → 9999
1270 → 4444
1271 → 4444
1272 → 888888
1273 → 4444
1274 → 4444

Then all at once, a number-ticket turns golden and the rollercoaster-ride doesn’t end. So far, at least. I’ve tried, but I haven’t been able to find a rep-digit for 3515 and 3529 = 3515+digsum(3515) and so on:


3509 → 4444
3510 → 9999
3511 → 4444
3512 → 4444
3513 → 888888
3514 → 4444
3515 → ?
3516 → 888888
3517 → 4444
3518 → 4444
3519 → 9999
3520 → 4444
3521 → 4444
3522 → 888888
3523 → 4444
3524 → 4444
3525 → 888888
3526 → 4444
3527 → 4444
3528 → 9999
3529 → ?
3530 → 4444
3531 → 888888
3532 → 4444

Does 3515 ever yield a rep-digit for n += digsum(n)? It’s hard to believe it doesn’t, but I’ve no idea how to prove that it does. Except by simply riding the rollercoaster. And if the ride with the 3515-ticket never reaches a rep-digit, the rollercoaster will never let you know. How could it?

But here’s an example in base 23 of how a ticket for n+1 can give you a dramatically longer ride than a ticket for n and n+2:


MI → EEE (524 → 7742)
MJ → EEE (525 → 7742)
MK → 444 (526 → 2212)
ML → 444 (527 → 2212)
MM → MMMMMM (528 → 148035888)
100 → 444 (529 → 2212)
101 → 444 (530 → 2212)
102 → EEE (531 → 7742)
103 → 444 (532 → 2212)
104 → 444 (533 → 2212)
105 → EEE (534 → 7742)
106 → EEE (535 → 7742)
107 → 444 (536 → 2212)
108 → EEE (537 → 7742)
109 → 444 (538 → 2212)
10A → MMMMMM (539 → 148035888)
10B → EEE (540 → 7742)
10C → EEE (541 → 7742)
10D → EEE (542 → 7742)
10E → EEE (543 → 7742)
10F → 444 (544 → 2212)
10G → EEE (545 → 7742)
10H → EEE (546 → 7742)
10I → EEE (547 → 7742)
10J → 444 (548 → 2212)
10K → 444 (549 → 2212)
10L → MMMMMM (550 → 148035888)
10M → EEE (551 → 7742)
110 → EEE (552 → 7742)

DeVil to Power

666 is the Number of the Beast described in the Book of Revelation:

13:18 Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.

But 666 is not just diabolic: it’s narcissistic too. That is, it mirrors itself using arithmetic, like this:

666^47 =

5,049,969,684,420,796,753,173,148,798,405,
  564,772,941,516,295,265,408,188,117,632,
  668,936,540,446,616,033,068,653,028,889,
  892,718,859,670,297,563,286,219,594,665,
  904,733,945,856 → 5 + 0 + 4 + 9 + 9 + 6 + 9 + 6 + 8 + 4 + 4 + 2 + 0 + 7 + 9 + 6 + 7 + 5 + 3 + 1 + 7 + 3 + 1 + 4 + 8 + 7 + 9 + 8 + 4 + 0 + 5 + 5 + 6 + 4 + 7 + 7 + 2 + 9 + 4 + 1 + 5 + 1 + 6 + 2 + 9 + 5 + 2 + 6 + 5 + 4 + 0 + 8 + 1 + 8 + 8 + 1 + 1 + 7 + 6 + 3 + 2 + 6 + 6 + 8 + 9 + 3 + 6 + 5 + 4 + 0 + 4 + 4 + 6 + 6 + 1 + 6 + 0 + 3 + 3 + 0 + 6 + 8 + 6 + 5 + 3 + 0 + 2 + 8 + 8 + 8 + 9 + 8 + 9 + 2 + 7 + 1 + 8 + 8 + 5 + 9 + 6 + 7 + 0 + 2 + 9 + 7 + 5 + 6 + 3 + 2 + 8 + 6 + 2 + 1 + 9 + 5 + 9 + 4 + 6 + 6 + 5 + 9 + 0 + 4 + 7 + 3 + 3 + 9 + 4 + 5 + 8 + 5 + 6 = 666

666^51 =

993,540,757,591,385,940,334,263,511,341,
295,980,723,858,637,469,431,008,997,120,
691,313,460,713,282,967,582,530,234,558,
214,918,480,960,748,972,838,900,637,634,
215,694,097,683,599,029,436,416 → 9 + 9 + 3 + 5 + 4 + 0 + 7 + 5 + 7 + 5 + 9 + 1 + 3 + 8 + 5 + 9 + 4 + 0 + 3 + 3 + 4 + 2 + 6 + 3 + 5 + 1 + 1 + 3 + 4 + 1 + 2 + 9 + 5 + 9 + 8 + 0 + 7 + 2 + 3 + 8 + 5 + 8 + 6 + 3 + 7 + 4 + 6 + 9 + 4 + 3 + 1 + 0 + 0 + 8 + 9 + 9 + 7 + 1 + 2 + 0 + 6 + 9 + 1 + 3 + 1 + 3 + 4 + 6 + 0 + 7 + 1 + 3 + 2 + 8 + 2 + 9 + 6 + 7 + 5 + 8 + 2 + 5 + 3 + 0 + 2 + 3 + 4 + 5 + 5 + 8 + 2 + 1 + 4 + 9 + 1 + 8 + 4 + 8 + 0 + 9 + 6 + 0 + 7 + 4 + 8 + 9 + 7 + 2 + 8 + 3 + 8 + 9 + 0 + 0 + 6 + 3 + 7 + 6 + 3 + 4 + 2 + 1 + 5 + 6 + 9 + 4 + 0 + 9 + 7 + 6 + 8 + 3 + 5 + 9 + 9 + 0 + 2 + 9 + 4 + 3 + 6 + 4 + 1 + 6 = 666

But those are tiny numbers compared to 6^(6^6). That means 6^46,656 and equals roughly 2·6591… x 10^36,305. It’s 36,306 digits long and its full digit-sum is 162,828. However, 666 lies concealed in those digits too. To see how, consider the function Σ(x1,xn), which returns the sum of digits 1 to n of x. For example, π = 3·14159265…, so Σ(π14) = 3 + 1 + 4 + 1 = 9. The first 150 digits of 6^(6^6) are these:

26591197721532267796824894043879185949053422002699
24300660432789497073559873882909121342292906175583
03244068282650672342560163577559027938964261261109
… (150 digits)

If x = 6^(6^6), then Σ(x1,x146) = 666, Σ(x2,x148) = 666, and Σ(x2,x149) = 666.

There’s nothing special about these patterns: infinitely many numbers are narcissistic in similar ways. However, 666 has a special cultural significance, so people pay it more attention and look for patterns related to it more carefully. Who cares, for example, that 667 = digit-sum(667^48) = digit-sum(667^54) = digit-sum(667^58)? Fans of recreational maths will, but not very much. The Number of the Beast is much more fun, narcissistically and otherwise:

666 = digit-sum(6^194)
666 = digit-sum(6^197)

666 = digit-sum(111^73)
666 = digit-sum(111^80)

666 = digit-sum(222^63)
666 = digit-sum(222^66)

666 = digit-sum(333^58)
666 = digit-sum(444^53)
666 = digit-sum(777^49)
666 = digit-sum(999^49)


Previously pre-posted (please peruse):

More Narcissisum
Digital Disfunction
The Hill to Power
Narcissarithmetic #1
Narcissarithmetic #2