Sometimes patterns do go on forever, but sometimes apparent patterns fail.

Some fail quickly:

See SlicingTheCake

Some fail less quickly:

Here are the first few values ...
n k(n) Divides
1 0 Yes
2 2 Yes
3 3 Yes
4 2 No
5 5 Yes
6 5 No
7 7 Yes
8 10 No
n k(n) Divides
9 12 No
10 17 No
11 22 Yes
12 29 No
13 39 Yes
14 51 No
15 68 No
16 90 No
n k(n) Divides
17 119 Yes
18 158 No
19 209 Yes
20 277 No
21 367 No
22 486 No
23 644 Yes
24 853 No


Some fail even more slowly

For each number, colour it black if it has an odd number of prime number factors, and red if it has an even number of prime number factors. Count each factor each time it appears, so 12 has an odd number of prime factors, 2, 2 and 3

Now start from 2 and count +1 for each black number and -1 for each red number. It seems that the blacks are always ahead.

Number 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
Factors 1 1 2 1 2 1 3 2 2 1 3 1 2 2 4 1 3 1 3 ...
"Sign" + + - + - + + - - + + + - - - + + + + ...
Sum 1 2 1 2 1 2 3 2 1 2 3 4 3 2 1 2 3 4 5 ...

Are they always?


Some fail astonishingly slowly:


Discussion

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