Category Archives: Base Behavior

Strartifacts

by in Arithmetic, Base Behavior, Geometry, Integer Sequences, Mathematics and tagged , , , , , , , , | Leave a comment

Here’s a sequence of decreasing numbers. Which number comes next?

612 → 600 → 594 → 414 → 398 → 182 → ?

It’s 166, because the numbers decrease by the product of their digits higher than 0:

612 – 6*2 = 612 – 12 = 600 → 600 – 6 = 594 → 594 – 5*9*4 = 594 – 180 = 414 → 414 – 4*4 = 414 – 16 = 398 → 398 – 3*9*8 = 398 – 216 = 166

Eventually the sequence will reached 0 and stop. If you want to see how this function looks on a graph, here it is:

x = n <= 3722, f(i) -= digmul(f(i)) → 0 (click for larger)

The graph represents n on the x-axis, with the red circles marking n = 100 and n = 1000. The sequence of falling digit-products is on the y-axis, but the graph has a special feature there. The y-axis is compressed according to the size of n, so that n = 1000 falls to 0 with n -= digmul(n) in the same height as n = 100. Here’s a graph for the same function in base 7:

x = n <= 3722 in base 7, f(i) -= digmul(f(i)) → 0

Now the red circles represent 7^2 = 49, 7^3 = 343, 7^4 = 2401, i.e. 100b7, 1000b7, 10000b7. And you can try other functions for n = n – func(n) = n -= func(n). Here’s a graph for n -= hailstep(n), where hailstep(n) returns the number of steps in the Collatz sequence for n:

x = n <= 3722 in base 7, f(i) -= hailstep(f(i)) → f(i) < 2

You form a Collatz sequence by starting with a whole number and finding the next number according to two rules:

1. If n(i) is divisible by 2, n(i+1) = n(i) / 2
2. If n(i) is not divisible by 2, n(i+1) = n(i) * 3 + 1

So the Collatz sequence for n = 10 looks like this:

10 → 10 / 2 = 5 → 5 * 3 + 1 = 16 → 16 / 2 = 8 → 8 / 2 = 4 → 4 / 2 = 2 → 2 / 2 = 1.

When you reach 1, you stop. So that’s six steps for n = 10. But does every n reach 1 in the end? It’s a very simple question about a very simple function. But nobody knows and nobody can prove that either all numbers do or at least one number doesn’t. The German mathematician Lothar Collatz (1910-90) conjectured that all numbers do reach 1. But it can take a surprisingly long time, even with small n. This is the Collatz sequence for n = 27:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Now some more functions for the y-compressed fall-bands, as I call them. If you use the sum of the factors * powers, you get this:

x = n <= 7422, f(i) -= factpowsum(f(i)) → f(i) < 2

The factpowsum(n) is the sum of the factors multiplied by their powers. For example, 37692 = 2^2 * 3^3 * 349, so factpowsum(4188) = 2*2 + 3*3 + 349*1 = 362. Here’s factpowsum for more n:

x = n <= 14822, f(i) -= factpowsum(f(i)) → f(i) < 2

You can also use the very simple function f(i) -= 1, that is, compress the numbers from n to 1 into the y-gap. But if you do that, you’ll get a completely filled screen:

x = n <= 3722, f(i) -= 1 → f(i) = 0

So you can adjust the color of a pixel according to how many times it’s written to:

x = n <= 1862, f(i) -= 1 → f(i) = 0 (color-adjust)

The patterns in the colors are artifacts of the limited resolution of the screen, so I call these patterns strartifacts = strata + artifacts. Here’s another example:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (color-adjust)

Or adjust the greytone of the pixel:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (greytone-adjust)

And so on (in all cases, you can click for a larger image):

x = n <= 1862, f(i) -= blockmul(f(i)) (multiply run-lengths of same digits) → f(i) < 2

x is triangular(n) = 3 to 1734453, y is 1 < triangular numbers <= n

x = n <= , f(i) -= 1 → f(i) < 2

x = n <= 7442, f(i) -= 1 → f(i) < 2

x = n <= 1862, f(i) -= leaddig(f(i)) → f(i) < 2 # 1

x = n <= , f(i) -= leaddig(f(i)) → f(i) < 2 # 2

x = n <= , f(i) -= trailingdigit(f(i)) + 1 → f(i) < 2

x = n <= , f(i) -= trailingdigit(f(i) in base 5) + 3 → f(i) < 2

for triangular(n) = 3 to 6928503, f(i) -= primes → f(i) < 2


x = n <= 1862, f(i) -= blockmul(f(i) in base 5) (multiply run-lengths of same digits) → f(i) < 2

x = n <= 1862, f(i) -= blockmul(f(i) in base 2) → f(i) < 2

x = n <= 1862, f(i) -= digsum(f(i)) → f(i) < 2

x = n <= 3722, f(i) -= func(x = 1/4 → x < 0, x(1) = 4) → f(i) < 2

x = n <= 932, f(i) -= func(x -= 3/x → x < 0, x(1) = 6) → f(i) < 2

Fibonacci Friday Factors

by in Arithmetic, Base Behavior, Fibonacci Sequence, Integer Sequences, Mathematics, Ulam Spirals and tagged , , , , , , | Leave a comment

Today’s a Phiday Friday or Φiday Friday or Φriday, so let’s have some more Fibonacci Fun. Here is the famous Fibonacci sequence, where each number (after seeding with “0, 1”) is formed by adding the previous two numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, … — A000045 at the Online Encyclopedia of Integer Sequences (OEIS)

It’s obvious that the numbers get bigger for ever and that no number repeats except 1. But what happens to the final digit of the Fibonacci numbers, as underlined here?:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, …

If you think about it, you’ll realize that the final digit has to repeat. Look for the “0, 1, 1” re-appearing:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, … — A003893 at the OEIS, “a(n) = Fibonacci(n) mod 10”

As you’ll see, all the numbers 0 to 9 appear in that sequence. But what about the final two digits of the Fibonacci sequence? Do all the numbers 0 to 99 appear before the sequence repeats?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 44, 33, 77, 10, 87, 97, 84, 81, 65, 46, 11, 57, 68, 25, 93, 18, 11, 29, 40, 69, 9, 78, 87, 65, 52, 17, 69, 86, 55, 41, 96, 37, 33, 70, 3, 73, 76, 49, 25, 74, 99, 73, 72, 45, 17, 62, 79, 41, 20, 61, 81, 42, 23, 65, 88, 53, 41, 94, 35, 29, 64, … — A105471 at the OEIS, “a(n) = Fibonacci(n) mod 100”

And what about the the final three digits and the numbers 0 to 999?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 597, 584, 181, 765, 946, 711, 657, 368, 25, 393, 418, 811, 229, 40, 269, 309, 578, 887, 465, 352, 817, 169, 986, 155, 141, 296, 437, 733, 170, 903, 73, 976, 49, 25, 74, 99, 173, 272 … — A248740 at the OEIS, “a(n) = Fibonacci(n) mod 1000”

In fact, some numbers do go missing as the final block of digits gets longer. But all that is based on the representation of the Fibonacci numbers in base 10. What about other bases? I had a look at that question and came up with some interesting patterns when I represented the final-block numbers on an Ulam-like spiral, where numbers are represented as squares on a spiral rotating counter-clockwise. This is the spiral for powers of 2 (the red square marks the center of the spiral and the number 1):

Spiral of final Fib-digits modulo 2^p

Fib-spiral mod 2^p (smaller scale)

Fib-spiral mod 2^p (smaller scale still)

What fraction of numbers are missing from the spiral? Watch this space. In the meantime, here’s the Fib-spiral for powers of 3:

Fib-spiral mod 3^p

It’s completely filled, because no numbers are missing (the red square marks “1” at the center of the spiral). What about powers of 4? Well, we’ve already seen that Fib-spiral, because all powers of 4 are also powers of 2:

Fib-spiral mod 4^p

The Fib-spiral for powers of 5 is the same as the Fib-spiral for powers of 3: it’s completely filled again. But the Fib-spiral for powers of 6 is interesting:

Fib-spiral mod 6^p

Fib-spiral mod 6^p (smaller scale)

Fib-spiral mod 6^p (smaller scale still)

And here are more Fib-spirals and more interesting patterns:

Fib-spiral mod 7^p

Fib-spiral mod 10^p — identical to the Fib-spiral for 2^p

Fib-spiral mod 11^p

Fib-spiral mod 11^p (smaller scale)

Fib-spiral mod 13^p

Fib-spiral mod 13^p (smaller scale)

Fib-spiral mod 17^p

Fib-spiral mod 17^p (smaller scale)

Fib-spiral mod 19^p

Fib-spiral mod 19^p (smaller scale)

Fib-spiral mod 41^p

Fib-spiral mod 41^p (smaller scale)

Fib-spiral mod 47^p

Elsewhere Other-Accessible…

• Friday is Φday — a first look at Phiday on Friday

Run Fun

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

I was looking at numbers where the digit-runs of n in one base, b1, represented the digits of n in another base, b2. To understand digit-runs, take the number 775555111 (in whatever base). The function digitrun(775555111) = 2,4,3, because reading 775555111 left-to-right you see two 7s, four 5s, three 1s. Here’s one pattern I found:

10000011111 in base 2 = 1055 in base 10. The digit runs of 10000011111 are 1,5,5 and 15530 = 105510 = 100000111112.
10000022222 in base 3 = 59291 in base 10. The digit runs of 10000022222 are again 1,5,5 and 155241 = 5929110 = 100000222223.
1000009999910 → 1,5,5 and 15599998 = 1000009999910

The pattern isn’t difficult to understand — what are 30+2 and 99998+2? — and it’s easy to prove that it applies to all standard bases. But I wondered how AI would go about the proof. Here’s the result:

For base b ≥ 2, let the largest digit be d = b − 1. Then

100000dddddb

has digit runs

1, 5, 5

so the run-length encoding is always

155.

The value of 155 is interpreted in base

b5 − 2,

because

155b5−2 = (b5−2)2 + 5(b5−2) + 5.

Thus

100000dddddb = 155b5−2.

Base b n = 100000(b−1)(b−1)(b−1)(b−1)(b−1)b Run Lengths Compressed Form Base-10 Value of n
2 100000111112 1,5,5 15530 1,055
3 100000222223 1,5,5 155241 59,291
4 100000333334 1,5,5 1551022 1,049,599
5 100000444445 1,5,5 1553123 9,765,624
6 100000555556 1,5,5 1557774 60,466,175
7 100000666667 1,5,5 15516805 282,475,248
8 100000777778 1,5,5 15532766 1,074,266,111
9 100000888889 1,5,5 15559047 3,486,270,320
10 1000009999910 1,5,5 15599998 10,000,099,999
11 100000AAAAA11 1,5,5 155161049 26,094,908,050
12 100000BBBBB12 1,5,5 155248830 62,191,972,031
13 100000CCCCC13 1,5,5 155371291 136,858,491,292
14 100000DDDDD14 1,5,5 155537822 281,793,968,543
15 100000EEEEE15 1,5,5 155759373 547,918,611,874
16 100000FFFFF16 1,5,5 1551048574 1,099,512,627,199
17 100000GGGGG17 1,5,5 1551419855 2,141,423,647,200
18 100000HHHHH18 1,5,5 1551889566 4,058,363,974,751
19 100000IIIII19 1,5,5 1552476097 7,502,570,010,432
20 100000JJJJJ20 1,5,5 1553199998 13,120,000,000,000

(Here A = 10, B = 11, …, J = 19.)

Proof for All b ≥ 2

The number

100000(b−1)(b−1)(b−1)(b−1)(b−1)b

has value

b10 + (b−1)(b4 + b3 + b2 + b + 1).

Since

b4 + b3 + b2 + b + 1 = (b5 − 1)/(b − 1),

this becomes

b10 + (b5 − 1) = b10 + b5 − 1.

Now let

B = b5 − 2.

Then

155B = B2 + 5B + 5.

Substituting B = b5 − 2:

(B2 + 5B + 5) = (b5−2)2 + 5(b5−2) + 5

= b10−4b5+4+5b5−10+5

= b10 + b5−1.

Hence

b10 + b5 − 1 = (b5 − 2)2+5(b5 − 2)+5

for every b ≥ 2.

Therefore,

100000(b−1)5b = 155b5−2.

This is exactly the pattern observed in bases 2 to 20, and it holds for all bases.

Matchin’ Fraction

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

I wondered whether contfrac(a/b), the continued fraction of a/b, ever matched the digits of a and b in some base. The answer was yes. But I haven’t found any examples in base 10:

3,1,2 = contfrac(3/12) in base 9 = contfrac(3/11) in base 10
4,1,3 = contfrac(4/13) in b16 = contfrac(4/19) in b10
5,1,4 ← 5/14 in b25 = 5/29
6,1,5 ← 6/15 in b36 = 6/41
25,2 ← 2/52 in b9 = 2/47 → 23,2
7,1,6 ← 7/16 in b49 = 7/55
8,1,7 ← 8/17 in b64 = 8/71
9,1,8 ← 9/18 in b81 = 9/89
A,1,9 ← A/19 in b100 = 10/109 → 10,1,9
42,1,3 ← 4/213 in b8 = 4/139 → 34,1,3
4,1,2,3,3 ← 41/233 in b8 = 33/155
1,17,1,2,3 ← 117/123 in b14 = 217/227 → 1,21,1,2,3
3A,3 ← 3/A3 in b28 = 3/283 → 94,3
3,5,A,2 ← 35/A2 in b34 = 107/342 → 3,5,10,2
3,1,4,1,4,1,5 ← 314/1415 in b8 = 204/781
2,1,36,3,2 ← 213/632 in b12 = 303/902 → 2,1,42,3,2
3,2,11,2,2,2 ← 321/1222 in b9 = 262/911 → 3,2,10,2,2,2
4H,4 ← 4/H4 in b65 = 4/1109 → 277,4
6,2,1,3,J ← 62/13J in b35 = 212/1349 → 6,2,1,3,19
8,3,3,1,D ← 83/31D in b22 = 179/1487 → 8,3,3,1,13
93,1,8 ← 9/318 in b27 = 9/2222 → 246,1,8
1,3A,1,1,4,2,2 ← 13A1/1422 in b12 = 2281/2330 → 1,46,1,1,4,2,2
C7,1,B ← C/71B in b21 = 12/3119 → 259,1,11
1,2,2,1,O,F ← 122/1OF in b50 = 2602/3715 → 1,2,2,1,24,15
2,1,1,5,55 ← 211/555 in b28 = 1597/4065 → 2,1,1,5,145
3,1,1,A,K,6 ← 311/AK6 in b29 = 2553/8996 → 3,1,1,10,20,6
1,2[70],1,3,9 ← 12[70]/139 in b98 = 9870/9907 → 1,266,1,3,9
1,E,4,1,M,7 ← 1E4/1M7 in b100 = 11404/12207 → 1,14,4,1,22,7
LG,5,4 ← L/G54 in b28 = 21/12688 → 604,5,4
G4,1,F ← G/41F in b64 = 16/16463 → 1028,1,15

Powers of Persistence

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

“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)

Primal Pellicles

by in Arithmetic, Base Behavior, Integer Sequences, Mathematics, Prime numbers and tagged , , , , , , , , | Leave a comment

Numbers have thin skins. And they’re easily replaced. Take 71624133. Here it is permuting its pellicles:

71624133 in base 10 = 100010001001110010111000101 in base 2 = 11222202212211200 in b3 = 10101032113011 in b4 = 121313433013 in b5 = 11035053113 in b6 = 1526536500 in b7 = 421162705 in b8 = 158685750 in b9 = 374802A9 in b11 = 1BBA1199 in b12 = 11AB9B59 in b13 = 9726137 in b14 = 644BE73 in b15 = F3855B7 in b16

But if digits are the skin of 71624133, what are its bones? Well, you could say the skeleton of a number, something that doesn’t change from base to base, is its prime factorization:

71624133 = 32 × 72 × 162413

But the primes themselves are numbers, so they’re wearing pellicles too. And it turns out that, in base 10, the pellicles of the prime factors of 71624133 match the pellicle of 71624133 itself:

71624133 = 32.72.162413

Here’s a list of primal pellicles in base 10:

735 = 3.5.72
3792 = 24.3.79
1341275 = 52.13.4127
13115375 = 53.7.13.1153
22940075 = 52.229.4007
29373375 = 3.53.29.37.73
71624133 = 32.72.162413
311997175 = 52.7.172.31.199
319953792 = 27.3.53.79.199
1019127375 = 32.53.7.127.1019
1147983375 = 3.53.7.11.83.479
1734009275 = 52.173.400927
5581625072 = 24.5581.62507
7350032375 = 53.7.23.73.5003
17370159615 = 34.5.17.59.61.701
33061224492 = 22.33.306122449
103375535837 = 72.37.103.553583
171167303912 = 23.11.172.6730391
319383665913 = 3.133.19.383.6659
533671737975 = 34.52.17.53.367.797
2118067737975 = 32.52.7.79.211.80677
3111368374257 = 3.112.132.683.74257
3216177757191 = 3.73.191.757.21617
3740437158475 = 52.37.4043715847
3977292332775 = 3.52.292.233.277.977
4417149692375 = 53.7.23.4969.44171
7459655393232 = 24.32.72.23.45965539
7699132721175 = 3.52.72.27211.76991
7973529228735 = 3.5.7.972.2287.3529
10771673522535 = 34.5.67.71.107.52253

You can find them at the Online Encyclopedia of Integer Sequences under A121342, “Composite numbers that are a concatenation of their distinct prime divisors in some order.” But what about pairs of primal pellicles, that is, pairs of numbers where the prime factors of each form the pellicle of the other?

35 = 5.7 ↔ 75 = 3.52
1275 = 3.52.17 ↔ 3175 = 52.127
131715 = 32.5.2927 ↔ 329275 = 52.13171
3199767 = 3.359.2971 ↔ 35932971 = 3.19.67.972
14931092 = 22.11.61.5563 ↔ 116155632 = 24.3.109.1492

And here are a few primal pellicles I’ve found in other bases:

Primal Pellicles in Base 2

1111011011110 = 10.1110.110110111 in b2 = 7902 = 2.32.439 in b10
1110001100110111 = 1110.10111.100011001 in b2 = 58167 = 32.23.281 in b10
1111011011011110 = 10.1110.110110110111 in b2 = 63198 = 2.32.3511 in b10
11101001100001101 = 1110.101.101001100001 in b2 = 119565 = 32.5.2657 in b10
1111011011011011110 = 10.1110.110110110110111 in b2 = 505566 = 2.32.28087 in b10
1111011111101111011 = 1110.1011.10111.11011111 in b2 = 507771 = 32.11.23.223 in b10

Primal Pellicles in Base 3

121022 = 210.12.102 in b3 = 440 = 23.5.11 in b10
212212 = 22.21.212 in b3 = 644 = 22.7.23 in b10
20110112 = 210.201.1011 in b3 = 4712 = 23.19.31 in b10
21110110 = 10.212.1101 in b3 = 5439 = 3.72.37 in b10
121111101 = 122.111.1101 in b3 = 12025 = 52.13.37 in b10
222112121 = 22.21.221121 in b3 = 19348 = 22.7.691 in b10
2202122021 = 22.2021.22021 in b3 = 54412 = 22.61.223 in b10
120212201221 = 2.122.21.201.1202 in b3 = 312550 = 2.52.7.19.47 in b10

Primal Pellicles in Base 7

2525 = 2.52.25 in b7 = 950 = 2.52.19 in b10
3210 = 2.34.10 in b7 = 1134 = 2.34.7 in b10
5252 = 2.52.52 in b7 = 1850 = 2.52.37 in b10
332616 = 33.16.326 in b7 = 58617 = 33.13.167 in b10
336045 = 32.5.3604 in b7 = 59715 = 32.5.1327 in b10
2251635 = 22.3.5.16.252 in b7 = 281580 = 22.3.5.13.192 in b10

Primal Pellicles in Base 11

253 = 22.3.52 in b11 = 300 = 22.3.52 in b10
732 = 2.32.72 in b11 = 882 = 2.32.72 in b10
2123 = 23.33.12 in b11 = 2808 = 23.33.13 in b10
3432 = 25.3.43 in b11 = 4512 = 25.3.47 in b10
3710 = 32.72.10 in b11 = 4851 = 32.72.11 in b10
72252 = 23.72.225 in b11 = 105448 = 23.72.269 in b10

Primal Pellicles in Base 15

275 = 24.5.7 in b15 = 560 = 24.5.7 in b10
2D5 = 2.52.D in b15 = 650 = 2.52.13 in b10
2CD5 = 2.52.CD in b15 = 9650 = 2.52.193 in b10
7BE3 = 3.72.BE in b15 = 26313 = 3.72.179 in b10
21285 = 24.52.128 in b15 = 105200 = 24.52.263 in b10

Summer-Time Twos

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

I wondered how often the digits of n2 appeared in sum(n1,n2). For example:

17 → 117 = sum(9,17)
20 → 200 = sum(5,20); 204,4; 207,3; 209,2 (c=4)

As I looked at higher n2, I found that the 2-views continued:

63 → 363 = sum(58,63); 638,53; 1638,28; 1763,23; 1863,18 (c=5)
88 → 1288 = sum(73,88); 2788,48; 2881,46; 3388,33; 3880,9; 3888,8 (c=6)
200 → 20009 = sum(14,200); 20022,13; 20034,12; 20045,11; 20055,10; 20064,9;
20072,8; 20079,7; 20085,6; 20090,5; 20094,4; 20097,3; 20099,2 (c=13)
558 → 39558 = sum(483,558); 55833,448; 95583,348; 105558,318; 125580,247; 126558,243; 143558,158; 152558,83; 155583,28; 155808,18; 155825,17; 155841,16; 155856,15; 155870,14; 155883,13; 155895,12 (c=16)
2000 → 2000010 = sum(45,2000); 2000054,44; 2000097,43; 2000139,42; 2000180,41; 2000220,40; 2000259,39; 2000297,38; 2000334,37; 2000370,36; 2000405,35; 2000439,34; 2000472,33; 2000504,32; 2000535,31; 2000565,30;
2000594,29; 2000622,28; 2000649,27; 2000675,26; 2000700,25; 2000724,24; 2000747,23; 2000769,22; 2000790,21; 2000810,20; 2000829,19; 2000847,18; 2000864,17; 2000880,16; 2000895,15; 2000909,14; 2000922,13; 2000934,
12; 2000945,11; 2000955,10; 2000964,9; 2000972,8; 2000979,7; 2000985,6; 2000990,5; 2000994,4; 2000997,3; 2000999,2 (c=44)

But what about other bases?

Base 9

15 in b9 → 115 = sum(5,15) (n=14 in b10) (c=1)
18 in b9 → 118 = sum(11,17); 180,1 (n=17 in b10) (c=2)
20 in b9 → 203 = sum(4,18); 206,3; 208,2 (n=18 in b10) (c=3)
45 in b9 → 445 = sum(32,41); 745,25; 1045,15; 1145,5 (n=41 in b10) (c=4)
55 in b9 → 555 = sum(41,50); 1055,35; 1355,25; 1555,15; 1655,5 (n=50 in b10) (c=5)
65 in b9 → 665 = sum(50,59); 1265,45; 1665,35; 2065,25; 2265,15; 2365,5 (n=59 in b10) (c=6)
75 in b9 → 775 = sum(59,68); 1475,55; 2075,45; 2475,35; 2750,26; 2775,25; 3075,15; 3175,5 (n=68 in b10) (c=8)
85 in b9 → 885 = sum(68,77); 1685,65; 2385,55; 2885,45; 3385,35; 3685,25; 3853,17; 3885,15; 4085,5 (n=77 in b10) (c=9)
200 in b9 → 20003 = sum(13,162); 20016,13; 20028,12; 20040,11; 20050,10; 20058,8; 20066,7; 20073,6; 20078,5; 20083,4; 20086,3; 20088,2 (n=162 in b10) (c=12)
415 in b9 → 13415 = sum(311,338); 25415,345; 36415,315; 41525,302; 46415,275; 55415,245; 63415,215; 64155,212; 70415,175; 75415,145; 80415,115; 83415,75; 85415,45; 86415,15 (n=338 in b10) (c=14)
[…]
2000 in b9 → 2000028 = sum(38,1458); 2000070,41; 2000120,40; 2000158,38; 2000206,37; 2000243,36; 2000278,35; 2000323,34; 2000356,33; 2000388,32; 2000430,31; 2000460,30; 2000488,28; 2000526,27; 2000553,26; 2000578,25; 2000613,24; 2000636,23; 2000658,22; 2000680,21; 2000710,20; 2000728,18; 2000746,17; 2000763,16; 2000778,15; 2000803,14; 2000816,13; 2000828,12; 2000840,11; 2000850,10; 2000858,8; 2000866,7; 2000873,6; 2000878,5; 2000883,4; 2000886,3; 2000888,2 (n=1458 in b10) (c=37)

Base 11

16 in b11 → 116 = sum(6,16) (n=17 in b10) (c=1)
20 in b11 → 201 = sum(5,22); 205,4; 208,3; 20A,2 (n=22 in b10) (c=4)
56 in b11 → 556 = sum(50,61); 956,36; 1156,26; 1356,16; 1456,6 (n=61 in b10) (c=5)
66 in b11 → 666 = sum(61,72); 1066,46; 1466,36; 1669,2A; 1766,26; 1966,16; 1A66,6 (n=72 in b10) (c=7)
86 in b11 → 886 = sum(83,94); 1486,66; 1A86,56; 2486,46; 2886,36; 3086,26; 3286,16; 3386,6 (n=94 in b10) (c=8)
96 in b11 → 996 = sum(94,105); 1696,76; 2296,66; 2896,56; 3296,46; 3696,36; 3996,26; 4096,16; 4196,6 (n=105 in b10) (c=9)
A6 in b11 → AA6 = sum(105,116); 18A6,86; 25A6,76; 31A6,66; 37A6,56; 41A6,46; 45A6,36; 48A6,26; 4AA6,16; 50A6,6 (n=116 in b10) (c=10)
200 in b11 → 1200A = sum(156,242); 20001,15; 20015,14; 20028,13; 2003A,12; 20050,11; 20060,10; 2006A,A; 20078,9; 20085,8; 20091,7; 20097,6; 200A1,5; 200A5,4; 200A8,3; 200AA,2 (n=242 in b10) (c=16)
[…]
A66 in b11 → 1AA66 = sum(1260,1282); A1A66,966; 109A66,946; 182A66,866; 198A66,846; 23A666,786; 253A66,766; 267A66,746; 314A66,666; 326A66,646; 375A66,566; 385A66,546; 416A66,466; 424A66,446; 457A66,366; 463A66,346; 46A666,326; 488A66,266; 492A66,246; 4A6666,186; 4A9A66,166; 501A66,146; 50AA66,66; 510A66,46 (n=1282 in b10) (c=24)
2000 in b11 → 2000005 = sum(52,2662); 2000051,47; 2000097,46; 2000131,45; 2000175,44; 2000208,43; 200024A,42; 2000290,41; 2000320,40; 200035A,3A; 2000398,39; 2000425,38; 2000461,37; 2000497,36; 2000521,35; 2000555,34; 2000588,33; 200060A,32; 2000640,31; 2000670,30; 200069A,2A; 2000718,29; 2000745,28; 2000771,27; 2000797,26; 2000811,25; 2000835,24; 2000858,23; 200087A,22; 20008A0,21; 2000910,20; 200092A,1A; 2000948,19; 2000965,18; 2000981,17; 2000997,16; 2000A01,15; 2000A15,14; 2000A28,13; 2000A3A,12; 2000A50,11; 2000A60,10; 2000A6A,A; 2000A78,9; 2000A85,8; 2000A91,7; 2000A97,6; 2000AA1,5; 2000AA5,4; 2000AA8,3; 2000AAA,2 (n=2662 in b10) (c=51)

Base 3

12 in b3 → 112 = sum(2,12); 120,1 (n=5 in b10) (c=2)
20 in b3 → 120 = sum(4,6); 200,10; 202,2 (n=6 in b10) (c=3)
122 in b3 → 10122 = sum(11,17); 11122,22; 11220,21; 12122,2; 12200,1 (n=17 in b10) (c=5)
1212 in b3 → 121212 = sum(41,50); 1001212,1012; 1101212,212; 1112120,200; 1121212,112; 1201212,12 (n=50 in b10) (c=6)
1222 in b3 → 122222 = sum(44,53); 1101222,1002; 1111222,222; 1112220,221; 1212222,102; 1221222,2; 1222000,1 (n=53 in b10) (c=7)
2000 in b3 → 1112000 = sum(28,54); 1120000,1000; 2000020,21; 2000110,20; 2000122,12; 2000210,11; 2000220,10; 2000222,2 (n=54 in b10) (c=8)
[…]
20000 in b3 → 111120000 = sum(82,162); 111200000,10000; 200000010,111; 200000120,110; 200000222,102; 200001100,101; 200001200,100; 200001222,22; 200002020,21; 200002110,20; 200002122,12; 200002210,11; 200002220,10; 200002222,2 (n=162 in b10) (c=14)

Base 4

13 in b4 → 130 = sum(1,13) (n=7 in b10) (c=1)
20 in b4 → 201 = sum(3,8); 203,2 (n=8 in b10) (c=2)
200 in b4 → 20001 = sum(6,32); 20012,11; 20022,10; 20031,3; 20033,2 (n=32 in b10) (c=5)
2000 in b4 → 2000021 = sum(11,128); 2000103,22; 2000130,21; 2000210,20; 2000223,13; 2000301,12; 2000312,11; 2000322,10; 2000331,3; 2000333,2 (n=128 in b10) (c=10)
20000 in b4 → 200000003 = sum(23,512); 200000121,112; 200000232,111; 200001002,110; 200001111,103; 200001213,102; 200001320,101; 200002020,100; 200002113,33; 200002211,32; 200002302,31; 200002332,30; 200003021,23; 200003103,22; 200003130,21; 200003210,20; 200003223,13; 200003301,12; 200003312,11; 200003322,10; 200003331,3; 200003333,2 (n=512 in b10) (c=22)

Base 8

17 in b8 → 170 = sum(1,17) (n=15 in b10) (c=1)
20 in b8 → 202 = sum(4,16); 205,3; 207,2 (n=16 in b10) (c=3)
200 in b8 → 20011 = sum(11,128); 20023,12; 20034,11; 20044,10; 20053,7; 20061,6; 20066,5; 20072,4; 20075,3; 20077,2 (n=128 in b10) (c=10)
2000 in b8 → 2000020 = sum(32,1024); 2000057,37; 2000115,36; 2000152,35; 2000206,34; 2000241,33; 2000273,32; 2000324,31; 2000354,30; 2000403,27; 2000431,26; 2000456,25; 2000502,24; 2000525,23; 2000547,22; 200057
0,21; 2000610,20; 2000627,17; 2000645,16; 2000662,15; 2000676,14; 2000711,13; 2000723,12; 2000734,11; 2000744,10; 2000753,7; 2000761,6; 2000766,5; 2000772,4; 2000775,3; 2000777,2 (n=1024 in b10) (c=31)

Base 16

1F in b16 → 1F0 = sum(1,1F) (n=31 in b10) (c=1)
20 in b16 → 201 = sum(6,32); 206,5; 20A,4; 20D,3; 20F,2 (n=32 in b10) (c=5)
200 in b16 → 20003 = sum(23,512); 20019,16; 2002E,15; 20042,14; 20055,13; 20067,12; 20078,11; 20088,10; 20097,F; 200A5,E; 200B2,D; 200BE,C; 200C9,B; 200D3,A; 200DC,9; 200E4,8; 200EB,7; 200F1,6; 20
0F6,5; 200FA,4; 200FD,3; 200FF,2 (n=512 in b10) (c=22)
[…]
EE4 in b16 → 42EE4A = sum(961,EE4); 6EE413,16; 6EE428,15; 6EE43C,14; 6EE44F,13; 6EE461,12; 6EE472,11; 6EE482,10; 6EE491,F; 6EE49F,E; 6EE4AC,D; 6EE4B8,C; 6EE4C3,B; 6EE4CD,A; 6EE4D6,9; 6EE4DE,8; 6EE4E5,7; 6EE4EB,6; 6EE4F0,5; 6EE4F4,4; 6EE4F7,3; 6EE4F9,2; 6EE4FA,1 (n=3812 in b10) (c=23)
2000 in b16 → 2000001 = sum(5B,2000); 200005B,5A; 20000B4,59; 200010C,58; 2000163,57; 20001B9,56; 200020E,55; 2000262,54; 20002B5,53; 2000307,52; 2000358,51; 20003A8,50; 20003F7,4F; 2000445,4E; 2000492,4D; 20004DE,4C; 2000529,4B; 2000573,4A; 20005BC,49; 2000604,48; 200064B,47; 2000691,46; 20006D6,45; 200071A,44; 200075D,43; 200079F,42; 20007E0,41; 2000820,40; 200085F,3F; 200089D,3E; 20008DA,3D; 2000916,3C; 2000951,3B; 200098B,3A; 20009C4,39; 20009FC,38; 2000A33,37; 2000A69,36; 2000A9E,35; 2000AD2,34; 2000B05,33; 2000B37,32; 2000B68,31; 2000B98,30; 2000BC7,2F; 2000BF5,2E; 2000C22,2D; 2000C4E,2C; 2000C79,2B; 2000CA3,2A; 2000CCC,29; 2000CF4,28; 2000D1B,27; 2000D41,26; 2000D66,25; 2000D8A,24; 2000DAD,23; 2000DCF,22; 2000DF0,21; 2000E10,20; 2000E2F,1F; 2000E4D,1E; 2000E6A,1D; 2000E86,1C; 2000EA1,1B; 2000EBB,1A; 2000ED4,19; 2000EEC,18; 2000F03,17; 2000F19,16; 2000F2E,15; 2000F42,14; 2000F55,13; 2000F67,12; 2000F78,11; 2000F88,10; 2000F97,F; 2000FA5,E; 2000FB2,D; 2000FBE,C; 2000FC9,B; 2000FD3,A; 2000FDC,9; 2000FE4,8; 2000FEB,7; 2000FF1,6; 2000FF6,5; 2000FFA,4; 2000FFD,3; 2000FFF,2 (n=8192 in b10) (c=90)

Previously Pre-Posted (Please Peruse)

• Summer-Time Hues
• Summer-Climb Views
• Summult-Time Hues

Feral Fractions

by in 666, Arithmetic, Base Behavior, Egyptian Fractions, Integer Sequences, Mathematics, Number of the Beast and tagged , , , , , | Leave a comment

“The uniquely unrepresentative ‘Egyptian’ fraction.” That’s what David Wells calls 2/3 = 0·666… in The Penguin Dictionary of Curious and Interesting Numbers (1986). Why unrepresentative”? Wells goes on to explain: “the Egyptians used only unit fractions, with this one exception. All other fractional quantities were expressed as sums of unit fractions.”

A unit fraction is 1 divided by a higher integer: 1/2, 1/3, 1/4, 1/5 and so on. Modern mathematicians are interested in those sums of unit fractions that produce integers, like this:

1 = 1/2 + 1/3 + 1/6 = egypt(2,3,6)
1 = 1/2 + 1/4 + 1/6 + 1/12 = egypt(2,4,6,12)
1 = 1/2 + 1/3 + 1/10 + 1/15 = egypt(2,3,10,15)
1 = egypt(2,4,10,12,15)
1 = egypt(3,4,6,10,12,15)
1 = egypt(2,3,9,18)
1 = egypt(2,4,9,12,18)
1 = egypt(3,4,6,9,12,18)
1 = egypt(2,6,9,10,15,18)
1 = egypt(3,4,9,10,12,15,18)
1 = egypt(2,4,5,20)
1 = egypt(3,4,5,6,20)
1 = egypt(2,5,6,12,20)
1 = egypt(3,4,5,10,15,20)
1 = egypt(2,5,10,12,15,20)
1 = egypt(3,5,6,10,12,15,20)
1 = egypt(3,4,5,9,18,20)
1 = egypt(2,5,9,12,18,20)
1 = egypt(3,5,6,9,12,18,20)
1 = egypt(4,5,6,9,10,15,18,20)

2 = egypt(2,3,4,5,6,8,9,10,15,18,20,24)
2 = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 + 1/24

Sums-to-integers like those are called Egyptian fractions, for short. I looked for some such sums that included 1/666:

1 = egypt(2,3,7,63,222,518,666)
1 = egypt(2,3,8,36,111,296,666)
1 = egypt(2,3,9,20,444,555,666)
1 = egypt(2,3,9,21,222,518,666)
1 = egypt(2,3,9,24,111,296,666)
1 = egypt(2,3,9,26,74,481,666)
1 = egypt(2,4,8,9,111,296,666)

And I looked for Egyptian fractions whose denominators summed to rep-digits like 111 and 666 (denominators are the bit below the stroke of 1/3 or 2/3, where the bit above is called the numerator):

1 = egypt(4,6,7,9,10,14,15,18,28)
111 = 4+6+7+9+10+14+15+18+28

1 = egypt(3,6,8,9,10,15,21,24,126)
222 = 3+6+8+9+10+15+21+24+126

1 = egypt(2,6,8,12,16,17,272)
333 = 2+6+8+12+16+17+272

1 = egypt(2,4,9,11,22,396)
444 = 2+4+9+11+22+396

1 = egypt(5,6,9,10,11,12,15,20,21,22,28,396)
555 = 5+6+9+10+11+12+15+20+21+22+28+396

1 = egypt(2,6,8,10,15,25,600)
666 = 2+6+8+10+15+25+600

1 = egypt(4,5,8,12,14,18,20,21,24,26,28,819)
999 = 4+5+8+12+14+18+20+21+24+26+28+819

Alas, Egyptian fractions like those are attractive but trivial. This isn’t trivial, though:

Prof Greg Martin of the University of British Columbia has found a remarkable Egyptian fraction for 1 with 454 denominators all less than 1000.

1 = egypt(97, 103, 109, 113, 127, 131, 137, 190, 192, 194, 195, 196, 198, 200, 203, 204, 205, 206, 207, 208, 209, 210, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 225, 228, 230, 231, 234, 235, 238, 240, 244, 245, 248, 252, 253, 254, 255, 256, 259, 264, 265, 266, 267, 268, 272, 273, 274, 275, 279, 280, 282, 284, 285, 286, 287, 290, 291, 294, 295, 296, 299, 300, 301, 303, 304, 306, 308, 309, 312, 315, 319, 320, 321, 322, 323, 327, 328, 329, 330, 332, 333, 335, 338, 339, 341, 342, 344, 345, 348, 351, 352, 354, 357, 360, 363, 364, 365, 366, 369, 370, 371, 372, 374, 376, 377, 378, 380, 385, 387, 390, 391, 392, 395, 396, 399, 402, 403, 404, 405, 406, 408, 410, 411, 412, 414, 415, 416, 418, 420, 423, 424, 425, 426, 427, 428, 429, 430, 432, 434, 435, 437, 438, 440, 442, 445, 448, 450, 451, 452, 455, 456, 459, 460, 462, 464, 465, 468, 469, 470, 472, 473, 474, 475, 476, 477, 480, 481, 483, 484, 485, 486, 488, 490, 492, 493, 494, 495, 496, 497, 498, 504, 505, 506, 507, 508, 510, 511, 513, 515, 516, 517, 520, 522, 524, 525, 527, 528, 530, 531, 532, 533, 536, 539, 540, 546, 548, 549, 550, 551, 552, 553, 555, 558, 559, 560, 561, 564, 567, 568, 570, 572, 574, 575, 576, 580, 581, 582, 583, 584, 585, 588, 589, 590, 594, 595, 598, 603, 605, 608, 609, 610, 611, 612, 616, 618, 620, 621, 623, 624, 627, 630, 635, 636, 637, 638, 640, 642, 644, 645, 646, 648, 649, 650, 651, 654, 657, 658, 660, 663, 664, 665, 666, 667, 670, 671, 672, 675, 676, 678, 679, 680, 682, 684, 685, 688, 689, 690, 693, 696, 700, 702, 703, 704, 705, 707, 708, 710, 711, 712, 713, 714, 715, 720, 725, 726, 728, 730, 731, 735, 736, 740, 741, 742, 744, 748, 752, 754, 756, 759, 760, 762, 763, 765, 767, 768, 770, 774, 775, 776, 777, 780, 781, 782, 783, 784, 786, 790, 791, 792, 793, 798, 799, 800, 804, 805, 806, 808, 810, 812, 814, 816, 817, 819, 824, 825, 826, 828, 830, 832, 833, 836, 837, 840, 847, 848, 850, 851, 852, 854, 855, 856, 858, 860, 864, 868, 869, 870, 871, 872, 873, 874, 876, 880, 882, 884, 888, 890, 891, 893, 896, 897, 899, 900, 901, 903, 904, 909, 910, 912, 913, 915, 917, 918, 920, 923, 924, 925, 928, 930, 931, 935, 936, 938, 940, 944, 945, 946, 948, 949, 950, 952, 954, 957, 960, 962, 963, 966, 968, 969, 972, 975, 976, 979, 980, 981, 986, 987, 988, 989, 990, 992, 994, 996, 999) — "Egyptian Fractions" by Ron Knott at Surrey University

Color-Coded ContFracs

by in Arithmetic, Base Behavior, Continued Fractions, Mathematics and tagged , , , , , | Leave a comment

Continued fractions are cool… Too cool for school. Or too cool for my school, at least. Because I never learnt about them there. Now that I have learnt about them, they’ve helped me wade a little further into the immeasurable Mare Mathematicum. Or Mare Matris Mathematicæ. I’m almost ankle-deep now, rather than just toe-deep. (I wish.)

But apart from aiding my understanding, continued fractions have always enhanced my entertainment. I can use them to find pretty but (probably) puny patterns like these:


[3,1,2] = contfrac(3/12) in base 9 = contfrac(3/11) in base 10
4,1,3 ← 4/13 in b16 = 4/19
5,1,4 ← 5/14 in b25 = 5/29
6,1,5 ← 6/15 in b36 = 6/41
7,1,6 ← 7/16 in b49 = 7/55
8,1,7 ← 8/17 in b64 = 8/71
9,1,8 ← 9/18 in b81 = 9/89
A,1,9 ← A/19 in b100 = 10/109 → 10,1,9
B,1,A ← B/1A in b121 = 11/131 → 11,1,10
C,1,B ← C/1B in b144 = 12/155 → 12,1,11

Those patterns with square numbers carry on for ever, I assume. I also assume that the similar patterns below do too, though I’m not sure if every base contains an infinite number of them. Maybe some bases don’t contain any at all. I haven’t found any in base 10 so far:


[25,2] = contfrac(2/52) in base 9 = contfrac(2/47) in base 10 = [23,2]
42,1,3 ← 4/213 in b8 = 4/139 → 34,1,3
4,1,2,3,3 ← 41/233 in b8 = 33/155
24,1,3,1,2 ← 24/1312 in b5 = 14/207 → 14,1,3,1,2
1,17,1,2,3 ← 117/123 in b14 = 217/227 → 1,21,1,2,3
320,1,2 ← 3/2012 in b5 = 3/257 → 85,1,2
254,2 ← 2/542 in b7 = 2/275 → 137,2
3A,3 ← 3/A3 in b28 = 3/283 → 94,3
3,5,A,2 ← 35/A2 in b34 = 107/342 → 3,5,10,2
12,1,5,3 ← 12/153 in b17 = 19/377 → 19,1,5,3
12,1,5,3 ← 12/153 in b17 = 19/377 → 19,1,5,3
3,1,4,1,4,1,5 ← 314/1415 in b8 = 204/781
2,1,36,3,2 ← 213/632 in b12 = 303/902 → 2,1,42,3,2
3,2,11,2,2,2 ← 321/1222 in b9 = 262/911 → 3,2,10,2,2,2
41,2,1,1,6 ← 41/2116 in b8 = 33/1102 → 33,2,1,1,6
4H,4 ← 4/H4 in b65 = 4/1109 → 277,4
249,2 ← 2/492 in b17 = 2/1311 → 655,2
6,2,1,3,J ← 62/13J in b35 = 212/1349 → 6,2,1,3,19
8,3,3,1,D ← 83/31D in b22 = 179/1487 → 8,3,3,1,13
142,1,1,6 ← 14/2116 in b9 = 13/1554 → 119,1,1,6
10,1,111,1,1,2 ← 1011/11112 in b6 = 223/1556 → 6,1,43,1,1,2
204,1,17 ← 20/4117 in b8 = 16/2127 → 132,1,15
93,1,8 ← 9/318 in b27 = 9/2222 → 246,1,8
1,3A,1,1,4,2,2 ← 13A1/1422 in b12 = 2281/2330 → 1,46,1,1,4,2,2
4340,1,3 ← 4/34013 in b5 = 4/2383 → 595,1,3
13,1,7,6 ← 13/176 in b46 = 49/2444 → 49,1,7,6
C7,1,B ← C/71B in b21 = 12/3119 → 259,1,11
35,3,2,3,1,1,2 ← 353/23112 in b6 = 141/3284 → 23,3,2,3,1,1,2
1,2,2,1,O,F ← 122/1OF in b50 = 2602/3715 → 1,2,2,1,24,15
2,1,1,5,55 ← 211/555 in b28 = 1597/4065 → 2,1,1,5,145
1P,2,H,2 ← 1P/2H2 in b47 = 72/5219 → 72,2,17,2
50,14,1,1,1,5 ← 501/41115 in b6 = 181/5447 → 30,10,1,1,1,5
5450,1,4 ← 5/45014 in b6 = 5/6274 → 1254,1,4
3103,1,2 ← 3/10312 in b9 = 3/6815 → 2271,1,2
4B,1,2,2,C ← 4B/122C in b19 = 87/7631 → 87,1,2,2,12
3G,D,2,3 ← 3G/D23 in b26 = 94/8843 → 94,13,2,3
3,1,1,A,K,6 ← 311/AK6 in b29 = 2553/8996 → 3,1,1,10,20,6
1,2[70],1,3,9 ← 12[70]/139 in b98 = 9870/9907 → 1,266,1,3,9
14,1,9,A ← 14/19A in b97 = 101/10292 → 101,1,9,10
14,1,9,A ← 14/19A in b97 = 101/10292 → 101,1,9,10
4133,1,14,2 ← 41/331142 in b5 = 21/11422 → 543,1,9,2
1,E,4,1,M,7 ← 1E4/1M7 in b100 = 11404/12207 → 1,14,4,1,22,7
LG,5,4 ← L/G54 in b28 = 21/12688 → 604,5,4

Matching Fractions

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

0.1666… = 1/6
0.0273972… = 2/73
0.0379746… = 3/79
0.0016181229… = 1/618
0.0027322404… = 2/732 → 1/366
0.0058548009… = 5/854
0.01393354769… = 13/933
0.07598784194… = 75/987 → 25/329
0.08998988877… = 89/989
0.141993957703… = 141/993 → 47/331
0.0005854115443… = 5/8541
0.00129282482223… = 12/9282 → 2/1547
0.00349722279366… = 34/9722 → 17/4861
0.013599274705349… = 135/9927 → 15/1103
0.0000273205382146… = 2/73205

0.0465103… = 4/65 in base 8 = 4/53 in base 10
0.13735223… = 13/73 in b8 = 11/59 in b10
0.0036256353… = 3/625 → 1/207 in b8 = 3/405 → 1/135 in b10
0.01172160236… = 11/721 → 3/233 in b8 = 9/465 → 3/155 in b10
0.01272533117… = 12/725 in b8 = 10/469 in b10
0.03175523464… = 31/755 in b8 = 25/493 in b10
0.06776766655… = 67/767 in b8 = 55/503 in b10
0.251775771755… = 251/775 in b8 = 169/509 in b10
0.0003625152504… = 3/6251 in b8 = 3/3241 in b10
0.00137303402723… = 13/7303 in b8 = 11/3779 in b10
0.00267525714052… = 26/7525 in b8 = 22/3925 in b10
0.035777577356673… = 357/7757 in b8 = 239/4079 in b10

0.3763… = 3/7 in b9 = 3/7 in b10
0.0155187… = 1/55 in b9 = 1/50 in b10
0.0371482… = 3/71 in b9 = 3/64 in b10
0.0474627… = 4/74 in b9 = 4/67 in b10
0.43878684… = 43/87 in b9 = 39/79 in b10
0.07887877766… = 78/878 in b9 = 71/719 in b10
0.01708848667… = 17/0884 → 4/221 in b9 = 16/724 → 4/181 in b10
0.170884866767… = 170/884 → 40/221 in b9 = 144/724 → 36/181 in b10

0.2828… = 2/8 → 1/4 in b11 = 2/8 → 1/4 in b10
0.4986… = 4/9 in b11 = 4/9 in b10
0.54A9A8A6… = 54/A9 in b11 = 59/119 in b10
0.0010A17039… = 1/A17 in b11 = 1/1228 in b10
0.010A170392A… = 10/A17 in b11 = 11/1228 in b10
0.01AA5854872… = 1A/A58 in b11 = 21/1273 in b10
0.027A716A416… = 27/A71 in b11 = 29/1288 in b10
0.032A78032A7… = 32/A78 → 1/34 in b11 = 35/1295 → 1/37 in b10
0.0190AA5A829… = 19/0AA5 → 4/221 in b11 = 20/1325 → 4/265 in b10
0.190AA5A829… = 190/AA5 → 40/221 in b11 = 220/1325 → 44/265 in b10

0.23B7A334… = 23/B7 in b12 = 27/139 in b10
0.075BA597224… = 75/BA5 in b12 = 89/1709 in b10
0.0ABBABAAA99… = AB/BAB in b12 = 131/1715 in b10
0.185BB5B859B4… = 185/BB5 in b12 = 245/1721 in b10