Category Archives: Arithmetic

Powers of Persistence

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“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)

Graph durch Euler

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

This is the famous Ulam spiral, in which prime numbers are represented on filled squares on a square spiral:

The Ulam spiral

I like the way the spiral sits between chaos and calm. It’s not wholly random and it’s not wholly regular — it’s betwixt and between. You get a similar chaos-and-calm vibe from a graph for a function called Euler phi. And primes are at work there too. Here’s the graph from Wikipedia:

Graph of eulerphi(n) = φ(n) (see Euler’s totient function)

But what is the Euler phi function? For any integer n, eulerphi(n) gives you the count of numbers < n that are relatively prime to n. That is, the count of numbers < n that have no common factors with n other than one. You can see how eulerphi(n) works by considering whether you can simplify the fraction a/b, where a = 1..n-1 and b = n:

φ(6) = 2
1/6 (1)
2/6 → 1/3
3/6 → 1/2
4/6 → 2/3
5/6, ∴ φ(6) = 2

φ(7) = 6
1/7 (1)
2/7 (2)
3/7 (3)
4/7 (4)
5/7 (5)
6/7, ∴ φ(7) = 6

φ(12) = 4
1/12 (1)
2/12 → 1/6
3/12 → 1/4
4/12 → 1/3
5/12 (2)
6/12 → 1/2
7/12 (3)
8/12 → 2/3
9/12 → 3/4
10/12 → 5/6
11/12, ∴ φ(12) = 4

φ(13) = 12
1/13 (1)
2/13 (2)
3/13 (3)
4/13 (4)
5/13 (5)
6/13 (6)
7/13 (7)
8/13 (8)
9/13 (9)
10/13 (10)
11/13 (11)
12/13, ∴ φ(13) = 12

As you can see, eulerphi(n) = n-1 for primes. Now you know what the top line of the Eulerphi graph is. It’s the primes. Here’s a bigger version of the graph:

Graph of eulerphi(n) = φ(n)

Unlike the Ulam spiral, however, the Eulerphi graph is cramped. But it’s easy to stretch it. You can represent φ(n) as a fraction between 0 and 1 like this: phifrac(n) = φ(n) / (n-1). Using phifrac(n), you can create Eulerphi bands, like this:

Eulerphi band, n <= 1781

Eulerphi band, n <= 3561

Eulerphi band, n <= 7121

Eulerphi band, n <= 14241

Or you can create Eulerphi discs, like this:

Eulerphi disc, n <= 1601

Eulerphi disc, n <= 3201

Eulerphi disc, n <= 6401

Eulerphi disc, n <= 12802

Eulerphi disc, n <= 25602

But what is the bottom line of the Eulerphi bands and inner ring of the Eulerphi discs, where φ(n) is smallest relative to n? Well, the top line or outer ring is the primes and the bottom line or inner ring is the primorials (and their multiples). The function primorial(n) is the multiple of the first n primes:

primorial(1) = 2
primorial(2) = 2*3 = 6
primorial(3) = 2*3*5 = 30
primorial(4) = 2*3*5*7 = 210
primorial(5) = 2*3*5*7*11 = 2310
primorial(6) = 2*3*5*7*11*13 = 30030
primorial(7) = 2*3*5*7*11*13*17 = 510510
primorial(8) = 2*3*5*7*11*13*17*19 = 9699690
primorial(9) = 2*3*5*7*11*13*17*19*23 = 223092870
primorial(10) = 2*3*5*7*11*13*17*19*23*29 = 6469693230

Here are the numbers returning record lows for φfrac(n) = φ(n) / (n-1):

φ(4) = 2 (2/3 = 0.666…)
4 = 2^2
φ(6) = 2 (2/5 = 0.4)
6 = 2.3
φ(12) = 4 (4/11 = 0.363636…)
12 = 2^2.3
[…]
φ(30) = 8 (8/29 = 0.275862…)
30 = 2.3.5
φ(60) = 16 (16/59 = 0.27118…)
60 = 2^2.3.5
[…]
φ(210) = 48 (48/209 = 0.229665…)
210 = 2.3.5.7
φ(420) = 96 (96/419 = 0.2291169…)
420 = 2^2.3.5.7
φ(630) = 144 (144/629 = 0.228934…)
630 = 2.3^2.5.7
[…]
φ(2310) = 480 (480/2309 = 0.2078822…)
2310 = 2.3.5.7.11
φ(4620) = 960 (960/4619 = 0.20783719…)
4620 = 2^2.3.5.7.11
[…]
30030 = 2.3.5.7.11.13
φ(60060) = 11520 (11520/60059 = 0.191811385…)
60060 = 2^2.3.5.7.11.13
φ(90090) = 17280 (17280/90089 = 0.1918103209…)
90090 = 2.3^2.5.7.11.13
[…]
φ(510510) = 92160 (92160/510509 = 0.18052571061…)
510510 = 2.3.5.7.11.13.17
φ(1021020) = 184320 (184320/1021019 = 0.18052553…)
1021020 = 2^2.3.5.7.11.13.17
φ(1531530) = 276480 (276480/1531529 = 0.180525474868579…)
1531530 = 2.3^2.5.7.11.13.17
φ(2042040) = 368640 (368640/2042039 = 0.18052544540040616…)
2042040 = 2^3.3.5.7.11.13.17

Primal Polynomial

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n² + n + 17 is one of the best-known polynomial formulas for primes. Its values for n = 0 to 15 are all prime, starting with 17 and ending with 257. — David Wells in The Penguin Dictionary of Curious and Interesting Numbers (1986), entry for “17”

• 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257

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

Lime Time

by in Arithmetic, Circles, Fractals, Geometry, Integer Sequences, Mathematics and tagged , , , , , , , , , , | Leave a comment

What do you get if you list every successive pair of entries in this sequence?

1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 1, 5, 2, 5, 3, 5, 4, 5, 1, 6, 5, 6, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 1, 8, 3, 8, 5, 8, 7, 8, 1, 9, 2, 9, 4, 9, 5, 9, 7, 9, 8, 9, 1, 10, 3, 10, 7, 10, 9, 10, 1, 11, 2, 11, 3, 11, 4, 11, 5, 11, 6, 11, 7, 11, 8, 11, 9, 11, 10, 11, 1, 12, 5, 12, 7, 12, 11, 12, 1, 13, … — A038568 at the Online Encyclopedia of Integer Sequence

You get the rational fractions ordered by denominator in their simplest form: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5… There are no pairs like 2/4 and 5/35, because those can be simplified: 2/4 → 1/2; 15/35 → 3/7. You can get the same set of rational fractions by listing every successive pair in this sequence, the Stern-Brocot sequence:

1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, … — A002487 at the OEIS

But the fractions don’t come ordered by denominator this time. In fact, they seem to come at random: 1/2, 1/3, 2/3, 1/4, 3/5, 2/5, 3/4, 1/5, 4/7, 3/8, 5/7, 2/7, 5/8… But they’re not random at all. There’s a complicated way of generating them and a simple way. An amazingly simple way, I think:

Moshe Newman proved that the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 – x). The successor function f(x) = 1/(floor(x) + 1 – frac(x)) can also be used. — A002487, “Stern-Brocot Sequence”, at the OEIS

In another form, the Stern-Brocot sequence is generated by what’s called the Calkin-Wilf Tree. Now suppose you use the Stern-Brocot sequence to supply the x co-ordinate of an L-graph whose arms run from 0 to 1. And you use the Calkin-Wilf Tree to supply the y co-ordinate of the L-tree. What do you get? As I described in “I Like Gryke”, you get this fractal:

Limestone fractal

I call it a limestone fractal or pavement fractal or gryke fractal, because it reminds me of the fissured patterns you see in the limestone pavements of the Yorkshire Dales:

Fissured limestone pavement, Yorkshire Dales (Wikipedia)

But what happens when you plot the (x,y) of the Stern-Brocot sequence and the Calkin-Wilf Tree on a circle instead? You get an interestingly distorted limestone fractal:

Limestone fractal on circle

You can also plot the (x,y) around the perimeter of a polygon, then stretch the polygon into a circle. Here’s a square:

Limestone fractal on square

⇓

Limestone square stretched to circle

And here are a pentagon, hexagon, heptagon and octagon — note the interesting perspective effects:

Limestone fractal on pentagon

⇓

Limestone pentagon stretched to circle

Limestone fractal on hexagon

⇓

Limestone hexagon stretched to circle

Limestone fractal on heptagon

⇓

Limestone heptagon stretched to circle

Limestone fractal on octagon

⇓

Limestone octagon stretched to circle

And finally, here are animations of limestone polygons stretching to circles:

Limestone square stretched to circle (animated at EZgif)

Limestone pentagon to circle (animated)

Limestone hexagon to circle (animated)

Limestone heptagon to circle (animated)

Limestone octagon to circle (animated)

Previously Pre-Posted (Please Peruse)

• I Like Gryke — a first look at the limestone fractal

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

Punctuated Pairimeters

by in Arithmetic, Circles, Geometry, Mathematics and tagged , , , , | Leave a comment

Imagine using the digits of n in two different bases to generate two fractions, a/b and c/d, where a/b < 1 and c/d < 1 (see Appendix for a sample program). Now use the fractions to find a pair of points on the perimeter of a circle, (x1, y1) and (x2, y2), then calculate and mark the midpoint of (x1, y1) and (x2, y2). If the bases have a prime factor in common, pretty patterns will appear from this punctuated pairimetry:

b1 = 2; b2 = 6

b1 = 2; b2 = 10

b1 = 2; b2 = 14

b1 = 4; b2 = 10

b1 = 4; b2 = 20

b1 = 4; b2 = 28

b1 = 6; b2 = 42

b1 = 12; b2 = 39

b1 = 24; b2 = 28

b1 = 28; b2 = 40

b1 = 32; b2 = 36

b1 = 42; b2 = 78

Appendix: Sample Program for Pairimetry

GetXY(xyi)=

fr = 0
recip = 1
bs = base[xyi]
for gi = 1 to di[xyi]
recip = recip/bs
fr += d[xyi,gi] * recip
next gi

x[xyi] = xcenter + sin(pi2 * fr) * radius
y[xyi] = ycenter + cos(pi2 * fr) * radius

endproc

Dinc(i1) =

d[i1,1]++;
if d[i1,1] == base[i1] then

i2 = 1

while d[i1,i2] == base[i1]

d[i1,i2] = 0
i2++;
d[i1,i2]++;

endwhile

if i2 > di[i1] then di[i1] = i2 endif

endif

endproc

Drawfigure =

base = x = y = di = array(2)
d = array(2,100)
radius = 100
pi2 = pi * 2
base[1] = 2
base[2] = 6
di[1] = 1
di[2] = 1

while true

for i = 1 to 2
call Dinc(i)
call GetXY(i)
next i

plot (x[1]+x[2]) / 2, (y[1] + y[2]) / 2

endwhile

endproc

call drawfigure

Summult-Time Hues

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

sum(3,6) = 3 * 6 = 18
• 3 * 2.3 = 2.3^2
sum(15,35) = 15 * 35 = 525
• 3.5 * 5.7 = 3.5^2.7
sum(85,204) = 85 * 204 = 17340
• 5.17 * 2^2.3.17 = 2^2.3.5.17^2
sum(493,1189) = 493 * 1189 = 586177
• 17.29 * 29.41 = 17.29^2.41
sum(2871,6930) = 2871 * 6930 = 19896030
• 3^2.11.29 * 2.3^2.5.7.11 = 2.3^4.5.7.11^2.29
sum(16731,40391) = 16731 * 40391 = 675781821
• 3^2.11.13^2 * 13^2.239 = 3^2.11.13^4.239
[…]

Elsewhere Other-Accessible

1, 18, 525, 17340, 586177, 19896030, 675781821, 22956120408, 779829016225, 26491211221770, 899921240562957, 30570830315362260, 1038508305678375841, 35278711540581704598, 1198437683944896688125, 40711602541832856049200, 1382996048733983114022337 — A011906 at the Online Encyclopedia of Integer Sequences

The Sumber of the B’s

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

First a bit of a boredom. Then a bit of beauty. These are the triangular numbers, including 666, the Number of the Beast:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, …

You can createthem as sumbers, that is, as numbers made by summing the whole numbers:

tri(1) = 1 = 1
tri(2) = 3 = 2+1
tri(3) = 6 = 3+2+1
tri(4) = 10 = 4+3+2+1
tri(5) = 15 = 5+4+3+2+1
tri(6) = 21 = 6+5+4+3+2+1
tri(7) = 28 = 7+6+5+4+3+2+1
tri(8) = 36 = 8+7+6+5+4+3+2+1
tri(9) = 45 = 9+8+7+6+5+4+3+2+1
tri(10) = 55 = 10+9+8+7+6+5+4+3+2+1

And here are the square numbers:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, …

You can create square numbers in various ways. Most obviously, by multiplying each whole number by itself:

sq(1) = 1*1 = 1
sq(2) = 2*2 = 4
sq(3) = 3*3 = 9
sq(4) = 4*4 = 16
sq(5) = 5*5 = 25
sq(6) = 6*6 = 36
sq(7) = 7*7 = 49
sq(8) = 8*8 = 64
sq(9) = 9*9 = 81
sq(10) = 10*10 = 100

Less obviously, by summing consecutive odd numbers:

sq(1) = 1 = 1
sq(2) = 1+3 = 4
sq(3) = 1+3+5 = 9
sq(4) = 1+3+5+7 = 16
sq(5) = 1+3+5+7+9 = 25
sq(6) = 1+3+5+7+9+11 = 36
sq(7) = 1+3+5+7+9+11+13 = 49
sq(8) = 1+3+5+7+9+11+13+15 = 64
sq(9) = 1+3+5+7+9+11+13+15+17 = 81
sq(10) = 1+3+5+7+9+11+13+15+17+19 = 100

And by summing pairs of consecutive triangular numbers (note that tri(0) = 0):

sq(1) = tri(0) + tri(1) = 0 + 1 = 1
sq(2) = tri(1) + tri(2) = 1 + 3 = 4
sq(3) = tri(2) + tri(3) = 3 + 6 = 9
sq(4) = tri(3) + tri(4) = 6 + 10 = 16
sq(5) = tri(4) + tri(5) = 10 + 15 = 25
sq(6) = tri(5) + tri(6) = 15 + 21 = 36
sq(7) = tri(6) + tri(7) = 21 + 28 = 49
sq(8) = tri(7) + tri(8) = 28 + 36 = 64
sq(9) = tri(8) + tri(9) = 36 + 45 = 81
sq(10) = tri(9) + tri(10) = 45 + 55 = 100

But sometimes squares are the sum of two triangular numbers that aren’t consecutive:

sq(4) = tri(1) + tri(5) = 1+15 = 16
sq(9) = tri(2) + tri(12) = 3+78 = 81
sq(16) = tri(2) + tri(22) = 3+253 = 256
sq(52) = tri(2) + tri(73) = 3+2701 = 2704
sq(14) = tri(3) + tri(19) = 6+190 = 196
sq(21) = tri(3) + tri(29) = 6+435 = 441
sq(44) = tri(9) + tri(61) = 45+1891 = 1936
sq(51) = tri(9) + tri(71) = 45+2556 = 2601
sq(49) = tri(10) + tri(68) = 55+2346 = 2401
sq(56) = tri(10) + tri(78) = 55+3081 = 3136
sq(16) = tri(11) + tri(19) = 66+190 = 256
sq(38) = tri(11) + tri(52) = 66+1378 = 1444
sq(54) = tri(11) + tri(75) = 66+2850 = 2916
sq(87) = tri(47) + tri(113) = 1128+6441 = 7569
sq(77) = tri(48) + tri(97) = 1176+4753 = 5929
sq(121) = tri(64) + tri(158) = 2080+12561 = 14641
sq(141) = tri(96) + tri(174) = 4656+15225 = 19881
sq(121) = tri(100) + tri(138) = 5050+9591 = 14641

Here’s a graph of squares that are the sum of any two triangular numbers, that is, is_square(tri(k1)+tri(k2)). The x axis is 1..k1 and the y axis is 1..k2, so the graph is symmetrical:

tri(k1) + tri(k2) = square(k3)

The (double) line at 45° represents squares that are the sum of consecutive triangulars. Other lines represent similarly regular patterns. Now for a bit of beauty. Things get more visually interesting when you test for squares that are the sums of any integer and a triangular number:

k1 + tri(k2) = square(k3)

The curves are optical oddities: where do they begin and end? The upper ones become lost to the eye in the lower ones. And vice versa. But you can force your eye to trace them further that it wants to.

Now try sums of integers and other polygonal numbers:

k1 + tri(k2) = pentagonal(k3)

k1 + square(k2) = pentagonal(k3)

k1 + pentagonal(k2) = square(k3)

k1 + hexagonal(k2) = pentagonal(k3)

And try other number sequences, like multiples of 4 with polygonals:

k1*4 + pentagonal(k2) = tri(k3)

k1*4 + square(k2) = tri(k3)

k1*4 + heptagonal(k2) = tri(k3)

And primes with polygonals:

tri(k1) + prime(k2) = tri(k3)

prime(k1) + tri(k2) = square(k3)

prime(k1) + octagonal(k2) = square(k3)

prime(k1) + pentagonal(k2) = square(k3)

prime(k1) + square(k2) = decagonal(k3)

prime(k1) + tri(k2) = hendecagonal(k3)

Partitional Pulchritude

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

If you want a good example of how, in math, something very simple can quickly get very deep, just look at partitions. Here are the partitions of 1 to 5, that is, the ways 1 to 5 can be expressed as a sum of integers smaller than or equal to themselves:

1 = 1

numbpart(1) = 1

2 = 2
1 + 1 = 2

numbpart(2) = 2

3 = 3
1 + 2 = 3
1 + 1 + 1 = 3

numbpart(3) = 3

4 = 4
1 + 3 = 4
2 + 2 = 4
1 + 1 + 2 = 4
1 + 1 + 1 + 1 = 4

numbpart(4) = 5

5 = 5
1 + 4 = 5
2 + 3 = 5
1 + 1 + 3 = 5
1 + 2 + 2 = 5
1 + 1 + 1 + 2 = 5
1 + 1 + 1 + 1 + 1 = 5

numbpart(5) = 7

It’s very easy to understand the concept of partitions, but very difficult to understand how partitions behave. For example, here is numbpart(n), the count of partitions for 1, 2, 3,…

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226, … A000041 at the Online Encyclopedia of Integer Sequences, “a(n) is the number of partitions of n (the partition numbers)”

What’s the formula for numbpart(n)? That’s a tricky question. And what’s the formula for the curves produced by counting the various lengths of partitions(n)? That’s another tricky question, but one thing is easy to see. As n gets bigger, the graph of countlen(partitions(n)) acquires a strange, lopsided beauty. Here are the partitions of 8, with the count of how many partitions of a particular length there are:

8 = 8 (1 partition of length 1)
1 + 7 = 8
2 + 6 = 8
3 + 5 = 8
4 + 4 = 8 (4 partitions of length 2)
1 + 1 + 6 = 8
1 + 2 + 5 = 8
1 + 3 + 4 = 8
2 + 2 + 4 = 8
2 + 3 + 3 = 8 (5 of length 3)
1 + 1 + 1 + 5 = 8
1 + 1 + 2 + 4 = 8
1 + 1 + 3 + 3 = 8
1 + 2 + 2 + 3 = 8
2 + 2 + 2 + 2 = 8 (5 of length 4)
1 + 1 + 1 + 1 + 4 = 8
1 + 1 + 1 + 2 + 3 = 8
1 + 1 + 2 + 2 + 2 = 8 (3 of length 5)
1 + 1 + 1 + 1 + 1 + 3 = 8
1 + 1 + 1 + 1 + 2 + 2 = 8 (2 of length 6)
1 + 1 + 1 + 1 + 1 + 1 + 2 = 8 (1 of length 7)
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 (1 of length 8)

When counts like that are shown as a graph, the graphs look like this (maximum counts are normalized to the same height):


graph of countlen(partitions(2))


countlen(partitions(3))


countlen(partitions(4))


countlen(partitions(5))


countlen(partitions(6))


countlen(partitions(7))


countlen(partitions(8))


countlen(partitions(9))


countlen(partitions(10))


countlen(partitions(15))


countlen(partitions(20))


countlen(partitions(30))


countlen(partitions(40))


countlen(partitions(50))


countlen(partitions(60))


countlen(partitions(70))


countlen(partitions(80))


countlen(partitions(90))


countlen(partitions(100))


Animated gif of partlen graphs (courtesy EZgif)

The graphs have a long, low right tail because the counts rise to great heights very quick, then fall away again, as you can see with partitions(100):

1 = count(partitions(10),len=1)
50 = count(partitions(10),len=2)
833 = count(partitions(10),len=3)
7153 = count(partitions(10),len=4)
38225 = count(partitions(10),len=5)
143247 = count(partitions(10),len=6)

[…]

10643083 = count(partitions(10),len=16)
11022546 = count(partitions(10),len=17)
11087828 = count(partitions(10),len=18)
10885999 = count(partitions(10),len=19)
10474462 = count(partitions(10),len=20)

[…]

30 = count(partitions(10),len=91)
22 = count(partitions(10),len=92)
15 = count(partitions(10),len=93)
11 = count(partitions(10),len=94)
7 = count(partitions(10),len=95)
5 = count(partitions(10),len=96)
3 = count(partitions(10),len=97)
2 = count(partitions(10),len=98)
1 = count(partitions(10),len=99)
1 = count(partitions(10),len=100)