I'm looking at the series generated by starting with n=1 and make r equal to the remainder of the sum of the digits of n when divided by 5, plus 1. Finally make n = n+r. So n goes:

1

then the sum of the digits of 1 is 1, and the remainder when divided by 5 is 1, plus 1 is 2. n becomes n+2 = 3.

3

7

10

12

the sum of the digits of 12 is 3 so the remainder of when divided by 5 is 3 plus 1 is 4, n+4 = 16.

16

19

20

23

24

...

[1, 3, 7, 10, 12, 16, 19, 20, 23, 24, 26, 30, 34, 37, 38, 40, 45, 50, 51, 53, 57, 60, 62, 66, 69, 70, 73, 74, 76, 80, 84, 87, 88, 90, 95, 100, 102, 106, 109, 110, 113, 114, 116, 120, 124, 127, 128, 130, 135, 140, 141, 143, 147, 150, 152, 156, 159, 160, 163, 164, 166, 170, 174, 177, 178, 180, 185, 190, 191, 193, 197, 200, 203, 204, 206, 210, 214, 217, 218, 220, 225, 230, 231, 233, 237, 240, 242, 246, 249, 250, 253, 254, 256, 260, 264, 267, 268, 270, 275, 280, 281]

Just looking at r each time gives:

[2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, "1", 2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, "2", 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, (2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, 2, 4, 3, 3), 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, 2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1]

See the first 18 numbers in r almost repeat but the 18th number is "1" and the 36th number is "2", then in parenthesis is the first 22 numbers but the last number is different.

These runs of repeating numbers get longer and longer but the series as a whole doesn't repeat. For example, looking at the first 51,000 numbers the first 1,000 repeat starting at:

repeats

1800,5040,8280,11520,13320,14760,16560,18000,19800,23040,26280,29520,31320,32760,34560,37440,40680,43920,45720,47160,48960,50400

There doesn't seem to be a consistent long range pattern of these numbers repeating. But there are many interesting patters, such as if we look at when the first 10,000 numbers repeat we find:

18000, 50400, 82800,... basically 10x as spread out as the repetition of 1000 numbers, so there is a sort of fractal arrangement of runs of numbers.

Of the first 1,000,000 numbers in the series, the distribution of 1's, 2's,3's, 4's, and 5's are:

[222,223, 222,222, 222,223, 222, 222, 111,110]

So there are almost exactly as many of every number but 5, which there are almost exactly half as many.

I say this series has low kolmogorov complexity because series with lots of runs of the same numbers can be written shortly by saying when these long runs occur and writing it once, and it works on every scale on this series, but I think it never does repeat exactly which makes it interesting as well.

Here's a picture of the first million terms of r, the first 1000 on the bottom row, the next 1000 on the next row up from left to right, with 1 as black, 2 as red, 3 as green, 4 as blue and 5 as white.

Zooming in somewhere near the middle:

and zooming about 8 times further in somewhere near the middle:

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