Tag Archives: math

On the M3!

by in Latin, Magistra Mundi, Mathematics and tagged , , , , , , , , , , , , , , , , | Leave a comment

6 = 2 x 3. And 6 = 1 + 2 + 3. But 6 also equals 3!. That is, 6 = 3 x 2 x 1, or factorial three. If you have three different items, you can arrange them in six different ways. There are three posibilities for the first item, two for the second and one for the third.

You can illustrate this linguistically. All languages are governed by mathematics, but maths manifests itself in different ways. Emphasis is an important part of language, for example, but there are different ways to achieve it. English usually does it with stress or by adding an emphatic word. Other languages can do it by varying the order of words. Latin, for example:

  • Mathematica Magistra Mundi
    — Mathematics is Mistress of the World.
  • Mathematica Mundi Magistra
    — Mathematics of the World is Mistress.
  • Magistra Mathematica Mundi
    — Mistress is Mathematics of the World
  • Magistra Mundi Mathematica
    — Mistress of the World is Mathematics.
  • Mundi Mathematica Magistra
    — Of the World Mathematics is Mistress.
  • Mundi Magistra Mathematica
    — Of the World the Mistress is Mathematics.

Elsewhere other-posted:

• Mathematica Magistra Mundi — more on the motto
• Moto-Motto — a variant on the motto

Prime Climb Time

by in Base Behavior, Binary numbers, Digit-sums, Mathematics, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The third prime is equal to the sum of the first and second primes: 2 + 3 = 5. After that, for obvious reasons, the prime-sum climbs much more rapidly than the primes themselves:

2, 3, 05, 07, 11, 13, 17, 19, 023, 029...
2, 5, 10, 17, 28, 41, 58, 77, 100, 129...

But what if you use digit-sum(p1..pn), i.e., the sum of the digits of the primes from the first to the nth? For example, the digit-sum(p1..p5) = 2 + 3 + 5 + 7 + 1+1 = 19, whereas the sum(p1..p5) = 2 + 3 + 5 + 7 + 11 = 28. Using the digit-sums of the primes, the comparison now looks like this:

2, 3, 05, 07, 11, 13, 17, 19, 23, 29...
2, 5, 10, 17, 19, 23, 31, 41, 46, 57...

The sum climbs more slowly, but still too fast. So what about a different base? In base-2, the digit-sum(p1..p3) = (1+0) + (1+1) + (1+0+1) = 1 + 2 + 2 = 5. The comparison looks like this:

2, 3, 05, 07, 11, 13, 17, 19, 23, 29...
1, 3, 05, 08, 11, 14, 16, 19, 23, 27...

For primes 3, 5, 11, 19, and 23, p = digit-sum(primes <= p) in base-2. But the cumulative digit-sum soon begins to climb too slowly:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271...

1, 3, 5, 8, 11, 14, 16, 19, 23, 27, 32, 35, 38, 42, 47, 51, 56, 61, 64, 68, 71, 76, 80, 84, 87, 091, 096, 101, 106, 110, 117, 120, 123, 127, 131, 136, 141, 145, 150, 155, 160, 165, 172, 175, 179, 184, 189, 196, 201, 206, 211, 218, 223, 230, 232, 236, 240, 245...

So what about base-3?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59...
2, 3, 6, 9, 12, 15, 20, 23, 28, 31, 34, 37, 42, 47, 52, 59, 64...

In base-3, for p = 2, 3 and 37, p = digit-sum(primes <= p), while for p = 23, 31, 47 and 59, p = digit-sum(primes < p), like this:

2 = 2.
3 = 2 + (1+0).
37 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3.

23 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3.
31 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3.
47 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) + (1+1+1+2) + (1+1+2+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3 + 5 + 5.
59 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) + (1+1+1+2) + (1+1+2+1) + (1+2+0+2) + (1+2+2+2) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3 + 5 + 5 + 5 + 7.

This carries on for a long time. For these primes, p = digit-sum(primes < p):

23, 31, 47, 59, 695689, 698471, 883517, 992609, 992737, 993037, 1314239, 1324361, 1324571, 1326511, 1327289, 1766291, 3174029

And for these primes, p = digit-sum(primes <= p):

3, 37, 695663, 695881, 1308731, 1308757, 1313153, 1314301, 1326097, 1766227, 3204779, 14328191

Now try the cumulative digit-sum in base-4:

2, 3, 5, 07, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59...
2, 5, 7, 11, 16, 20, 22, 26, 31, 36, 43, 47, 52, 59, 67, 72, 80...

The sum of digits climbs too fast. Base-3 is the Goldilocks base, climbing neither too slowly, like base-2, nor too fast, like all bases greater than 3.

Prime Time #2

by in Mathematics, Prime numbers and tagged , , , , , , , , , , , , , | Leave a comment

“2n2 + 29 is prime for all values of n for 1 to 28.” — The Penguin Dictionary of Curious and Interesting Numbers, David Wells (1986).

• 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597.

Elsewhere other-posted:

• Prime Time #1
• Poulet’s Propellor — Musings on Math and Mathculinity
• La Spirale è Mobile

Performativizing Papyrocentricity #22

by in Über-Transgression..., Book Reviews, Civilization, History, Literature, Mathematics, Philosophy of mathematics, Politics, The Sea and tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

• Plates from the Great – Shots from the Front: The British Soldier 1914-18, Richard Holmes (HarperPress 2008; paperback 2010)

• Math for the Mistress – A Mathematician’s Apology, G.H. Hardy (1940)

• Sinister Sinema – Scalarama: A Celebration of Subterranean Cinema at Its Sleazy, Slimy and Sinister Best, ed. Norman Foreman, B.A. (TransVisceral Books 2015)

• Rick Pickings – Lost, Stolen or Shredded: Stories of Missing Works of Art and Literature, Rick Gekoski (Profile Books 2013/2014)

• Slug is a Drug – Collins Complete Guide to British Coastal Wildlife, Paul Sterry and Andrew Cleave (HarperCollins 2012) (posted @ Overlord of the Über-Feral)

Or Read a Review at Random: RaRaR

He Say, He Sigh, He Sow #19

by in Magistra Mundi, Mathematics, Philosophy of mathematics, Quotations and tagged , , , , , , , , , , , , , , , , , | Leave a comment

“The study of mathematics is the indispensable basis for all intellectual and spiritual progress.” — F.M. Cornford (1874-1943) quoted in The Sacred in Music (see also Pythagoreanism).

This Mortal Doyle

by in A.E. Housman, Biology, Consciousness, Mathematics, Philosophy, Physics, Poetry, Science, Theology and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Challenger chopped and changed. That is to say, in one important respect, Arthur Conan Doyle’s character Professor Challenger lacked continuity. His philosophical views weren’t consistent. At one time he espoused materialism, at another he opposed it. He espoused it in “The Land of Mist” (1927):

“Don’t tell me, Daddy, that you with all your complex brain and wonderful self are a thing with no more life hereafter than a broken clock!”

“Four buckets of water and a bagful of salts,” said Challenger as he smilingly detached his daughter’s grip. “That’s your daddy, my lass, and you may as well reconcile your mind to it.”

But earlier, in “The Poison Belt” (1913), he had opposed it:

“No, Summerlee, I will have none of your materialism, for I, at least, am too great a thing to end in mere physical constituents, a packet of salts and three bucketfuls of water. Here ― here” ― and he beat his great head with his huge, hairy fist ― “there is something which uses matter, but is not of it ― something which might destroy death, but which death can never destroy.”

That story was published just over a century ago, but Challenger’s boast has not been vindicated in the meantime. So far as science can see, matter rules mind, not vice versa. Conan Doyle thought the same as the earlier Challenger, but Conan Doyle’s rich and teeming brain seems to have ended in “mere physical constituents”. To all appearances, when the organization of his brain broke down, so did his consciousness. And that concluded the cycle described by A.E. Housman in “Poem XXXII” of A Shropshire Lad (1896):

From far, from eve and morning
  And yon twelve-winded sky,
The stuff of life to knit me
  Blew hither: here am I.

Now – for a breath I tarry
  Nor yet disperse apart –
Take my hand quick and tell me,
  What have you in your heart.

Speak now, and I will answer;
  How shall I help you, say;
Ere to the wind’s twelve quarters
  I take my endless way. (ASL, XXXII)

Continue reading This Mortal Doyle…

Priamonds and Pearls

by in Mathematics, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Interesting patterns emerge when primes are represented as white blocks in a series of n-width left-right lines laid vertically, one atop the other. When the line is five blocks wide, the patterns look like this (the first green block is 1, followed by primes 2, 3 and 5, then 7 in the next line):
5line

(Click for larger version)

Right at the bottom of the first column is an isolated prime diamond, or priamond (marked with a green block). It consists of the four primes 307-311-313-317, where the three latter primes equal 307 + 4 and 6 and 10, or 307 + 5-1, 5+1 and 5×2 (the last prime in the first column is 331 and the first prime in the second is 337). About a third of the way down the first column is a double priamond, consisting of 97, 101, 103, 107, 109 and 113. For a given n, then, a priamond is a set of primes, p1, p2, p3 and p4, such that p2 = p1 + n-1, p3 = p + n+1 and p4 = p1 + 2n.

There are also fragments of pearl-necklace in the columns. One is above the isolated priamond. It consists of four prime-blocks slanting from left to right: 251-257-263-269, or 251 + 6, 12 and 18. A prearl-necklace, then, is a set of primes, p1, p2, p3…, such that p2 = p1 + n+i, p3 = p + 2(n+i)…, where i = +/-1. Now here are the 7-line and 9-line:

7line

Above: 7-line for primes

9line

Above: 9-line for primes

In the 9-line, you can see a prime-ladder marked with a red block. It consists of the primes 43-53-61-71-79-89-97-107, in alternate increments of 10 and 8, or 9+1 and 9-1. A prime-ladder, then, is a set of primes, p1, p2, p3, p4…, such that p2 = p1 + n+1, p3 = p + 2n, p3 = p + 3n+1…

And here is an animated gif of lines 5 through 51:

lines5to51

(Click or open in new window for larger version or if file fails to animate)

Miss This

by in Base Behavior, Mathematics, Narcissistic numbers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

1,729,404 is seven digits long. If you drop one digit at a time, you can create seven more numbers from it, each six digits long. If you add these numbers, something special happens:

1,729,404 → 729404 (missing 1) + 129404 (missing 7) + 179404 (missing 2) + 172404 + 172904 + 172944 + 172940 = 1,729,404

So 1,729,404 is narcissistic, or equal to some manipulation of its own digits. Searching for numbers like this might seem like a big task, but you can cut the search-time considerably by noting that the final two digits determine whether a number is a suitable candidate for testing. For example, what if a seven-digit number ends in …38? Then the final digit of the missing-digit sum will equal (3 x 1 + 8 x 6) modulo 10 = (3 + 48) mod 10 = 51 mod 10 = 1. This means that you don’t need to check any seven-digit number ending in …38.

But what about seven-digit numbers ending in …57? Now the final digit of the sum will equal (5 x 1 + 7 x 6) modulo 10 = (5 + 42) mod 10 = 47 mod 10 = 7. So seven-digit numbers ending in …57 are possible missing-digit narcissistic sums. Then you can test numbers ending …157, …257, …357 and so on, to determine the last-but-one digit of the sum. Using this method, one quickly finds the only two seven-digit numbers of this form in base-10:

1,729,404 → 729404 + 129404 + 179404 + 172404 + 172904 + 172944 + 172940 = 1,729,404

1,800,000 → 800000 + 100000 + 180000 + 180000 + 180000 + 180000 + 180000 = 1,800,000

What about eight-digit numbers? Only those ending in these two digits need to be checked: …00, …23, …28, …41, …46, …64, …69, …82, …87. Here are the results:

• 13,758,846 → 3758846 + 1758846 + 1358846 + 1378846 + 1375846 + 1375846 + 1375886 + 1375884 = 13,758,846
• 13,800,000 → 3800000 + 1800000 + 1300000 + 1380000 + 1380000 + 1380000 + 1380000 + 1380000 = 13,800,000
• 14,358,846 → 4358846 + 1358846 + 1458846 + 1438846 + 1435846 + 1435846 + 1435886 + 1435884 = 14,358,846
• 14,400,000 → 4400000 + 1400000 + 1400000 + 1440000 + 1440000 + 1440000 + 1440000 + 1440000 = 14,400,000
• 15,000,000 → 5000000 + 1000000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 = 15,000,000
• 28,758,846 → 8758846 + 2758846 + 2858846 + 2878846 + 2875846 + 2875846 + 2875886 + 2875884 = 28,758,846
• 28,800,000 → 8800000 + 2800000 + 2800000 + 2880000 + 2880000 + 2880000 + 2880000 + 2880000 = 28,800,000
• 29,358,846 → 9358846 + 2358846 + 2958846 + 2938846 + 2935846 + 2935846 + 2935886 + 2935884 = 29,358,846
• 29,400,000 → 9400000 + 2400000 + 2900000 + 2940000 + 2940000 + 2940000 + 2940000 + 2940000 = 29,400,000

But there are no nine-digit sumbers, or nine-digit numbers that supply missing-digit narcissistic sums. What about ten-digit sumbers? There are twenty-one:

1,107,488,889; 1,107,489,042; 1,111,088,889; 1,111,089,042; 3,277,800,000; 3,281,400,000; 4,388,888,889; 4,388,889,042; 4,392,488,889; 4,392,489,042; 4,500,000,000; 5,607,488,889; 5,607,489,042; 5,611,088,889; 5,611,089,042; 7,777,800,000; 7,781,400,000; 8,888,888,889; 8,888,889,042; 8,892,488,889; 8,892,489,042 (21 numbers)

Finally, the nine eleven-digit sumbers all take this form:

30,000,000,000 → 0000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 = 30,000,000,000

So that’s forty-one narcissistic sumbers in base-10. Not all of them are listed in Sequence A131639 at the Encyclopedia of Integer Sequences, but I think I’ve got my program working right. Other bases show similar patterns. Here are some missing-digit narcissistic sumbers in base-5:

• 1,243 → 243 + 143 + 123 + 124 = 1,243 (b=5) = 198 (b=10)
• 1,324 → 324 + 124 + 134 + 132 = 1,324 (b=5) = 214 (b=10)
• 1,331 → 331 + 131 + 131 + 133 = 1,331 (b=5) = 216 (b=10)
• 1,412 → 412 + 112 + 142 + 141 = 1,412 (b=5) = 232 (b=10)

• 100,000 → 00000 + 10000 + 10000 + 10000 + 10000 + 10000 = 100,000 (b=5) = 3,125 (b=10)
• 200,000 → 00000 + 20000 + 20000 + 20000 + 20000 + 20000 = 200,000 (b=5) = 6,250 (b=10)
• 300,000 → 00000 + 30000 + 30000 + 30000 + 30000 + 30000 = 300,000 (b=5) = 9,375 (b=10)
• 400,000 → 00000 + 40000 + 40000 + 40000 + 40000 + 40000 = 400,000 (b=5) = 12,500 (b=10)

And here are some sumbers in base-16:

5,4CD,111,0EE,EF0,542 = 4CD1110EEEF0542 + 5CD1110EEEF0542 + 54D1110EEEF0542 + 54C1110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD111EEEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEE0542 + 54CD1110EEEF542 + 54CD1110EEEF042 + 54CD1110EEEF052 + 54CD1110EEEF054 (b=16) = 6,110,559,033,837,421,890 (b=10)

6,5DD,E13,CEE,EF0,542 = 5DDE13CEEEF0542 + 6DDE13CEEEF0542 + 65DE13CEEEF0542 + 65DE13CEEEF0542 + 65DD13CEEEF0542 + 65DDE3CEEEF0542 + 65DDE1CEEEF0542 + 65DDE13EEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEE0542 + 65DDE13CEEEF542 + 65DDE13CEEEF042 + 65DDE13CEEEF052 + 65DDE13CEEEF054 (b=16) = 7,340,270,619,506,705,730 (b=10)

10,000,000,000,000,000 → 0000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 = 10,000,000,000,000,000 (b=16) = 18,446,744,073,709,551,616 (b=10)

F0,000,000,000,000,000 → 0000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 = F0,000,000,000,000,000 (b=16) = 276,701,161,105,643,274,240 (b=10)

Next I’d like to investigate sumbers created by missing two, three and more digits at a time. Here’s a taster:

1,043,101 → 43101 (missing 1 and 0) + 03101 (missing 1 and 4) + 04101 (missing 1 and 3) + 04301 + 04311 + 04310 + 13101 + 14101 + 14301 + 14311 + 14310 + 10101 + 10301 + 10311 + 10310 + 10401 + 10411 + 10410 + 10431 + 10430 + 10431 = 1,043,101 (b=5) = 18,526 (b=10)

Reverssum

by in Base Behavior, Digit-sums, Mathematics and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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 → [16] → 1D → DF → [17]3 → 4[18] → 107 → 709 → 914 → 424 → 42E → E35 → 54[17] → [17]5C → C7D → D96 → 6B3 → 3C7 → 7D6 → 6EE → E[16]2 → 2[18]8 → 90B → B1A → A2E → E3[17] → [17]5A → A7B → B90 → AC→ DD → F1 → 2C → C[16] → [18]2 → 40 → 8 (base=19)
• 1 → 2 → 4 → 8 → [16] → 1C → CE → F[18] → 108 → 80A → A16 → 627 → 731 → 13[18] → [18]43 → 363 → 36F → F77 → 794 → 4A7 → 7B5 → 5CA → ADC → CF5 → 5[17]4 → 4[18]B → B[19][17] → [18]1[18] → [18]3F → F5E → E79 → 994 → 4AB → BB9 → 9D2 → 2ED → DFB → B[17]C → C[19]B → C1E → E2[19] → [19]49 → 96B → B7F → F94 → 4B3 → 3C2 → 2D0 → D[17] → [19]3 → 51 → 1B → BD → EF → [17]3 → 4[17] → [18]5 → 71 → 1F → F[17] → [19]7 → 95 → 63 → 3F → [16]1 → 2D → D[17] (base=20)

Then it settles down again:

• 1 → 2 → 4 → 8 → [16] → 1B → BD → EE → [16]0 → 1B (base=21)
• 1 → 2 → 4 → 8 → [16] → 1A → AC → DA → BE → FE → [16]0 → 1A (base=22)
• 1 → 2 → 4 → 8 → [16] → 19 → 9B → C6 → 77 → 7[21] → [22]C → EA → BF → [16]E → [16]0 → 19 (base=23)

Base-33 is also short:

1 → 2 → 4 → 8 → [16] → [32] → 1[31] → [32]0 → 1[31] (base=33)

And so is base-35:

1 → 2 → 4 → 8 → [16] → [32] → 1[29] → [29][31] → [33][19] → [21]F → [16][22] → [23][19] → [20][30] → [32]0 → 1[29] (base=35)

So what about base-34?

1 → 2 → 4 → 8 → [16] → [32] → 1[30] → [30][32] → 10[24] → [24]0[26] → [26]26 → 63[26] → [26]47 → 75[29] → [29]6E → E8A → A9C → CA7 → 7B7 → 7B[32] → [32]C[23] → [23]E[31] → [31][16][23] → [23][18][33] → [33][20][29] → [29][23]D → D[25][26] → [26][27]9 → 9[29][20] → [20][30][33] → [33][33]1 → 21[32] → [32]23 → 341 → 14B → B4[17] → [17]59 → 96E → E74 → 485 → 58[21] → [21]95 → 5A[22] → [22]B8 → 8C[29] → [29]D[23] → [23]F[26] → [26][17][19] → [19][19][20] → [20][21]9 → 9[23]2 → 2[24]9 → 9[25]3 → 3[26]C → C[27]A → A[28][27] → [27][30]7 → 7[32][23] → [24]01 → 11F → F1[18] → [18]2F → F3[19] → [19]4[18] → [18]5[26] → [26]6[33] → [33]8[23] → [23]A[29] → [29]C[17] → [17]E[19] → [19]F[33] → [33][17][18] → [18][19][33] → [33][21][20] → [20][24]5 → 5[26]1 → 1[27]3 → 3[27][32] → [32][28][31] → [31][31][21] → [22]0C → C1[22] → [22]2D → D3[25] → [25]4[20] → [20]66 → 67[18] → [18]83 → 39D → D9[28] → [28]A[29] → [29]C[27] → [27]E[29] → [29][16][29] → [29][19]1 → 1[21]A → A[21][33] → [33][23]6 → 6[25][27] → [27][26][30] → [30][29]8 → 8[31][29] → [29][33]8 → 91[31] → [31]2[16] → [16]4C → C5E → E69 → 979 → 980 → 8[26] → [27]8 → 9[28] → [29]C → E2 → 2[30] → [31]0 → 1[28] → [28][30] → [32][18] → [20]E → F[20] → [21][16] → [17][24] → [25][24] → [26]6 → 7[24] → [25]4 → 5[20] → [20][30] → [32]2 → 3[32] → [33]4 → 62 → 2E → E[18] → [19]C → D[16] → [17]8 → 98 → 8[26] (1 leads to a 30-loop from 8[26] / 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