# [seqfan] Re: The Annoyance Sequence

Jay Anderson horndude77 at gmail.com
Fri Jul 21 07:22:04 CEST 2023

```On Thu, Jul 20, 2023 at 7:11 AM Allan Wechsler <acwacw at gmail.com> wrote:

> Could it be that 5 is the only number that never appears?
>

I believe 55 also gets in a loop. Here's the loop 5 gets is in:

initial state: 5, a, b, c, d, e, ... Where a, b, c, d, and e are all larger
than 5
annoy: a, b, c, d, e, 5, ...
pop: b, c, d, e, 5, ...
annoy: c, d, e, 5, ...
pop: d, e, 5, ...
annoy: e, 5, ...
pop: 5, ...

This last state is the same as the first with the next step to 'annoy'. So
5 definitely never appears. Given the above procedure it loops through
positions 1, 5, 3.

I ran the sequence up to 100,000 values. The smallest 3 values that don't
appear are 55, 1189, and 15413. 55 looks like it's getting in a similar
loop. It goes through positions 55 54 52 50 49 47 45 44 42 40 39 37 35 34
32 30 29 27 25 24 22 20 19 17 15 14 12 10 9 7 5 3 1. Each time it moves by
one is 5 below it being 'annoyed' -- every 3 steps. Near the end of the
loop it just so happens to line up with 5 so the loop repeats:

Initial: 5, _, _, _, 55
annoy: _, _, _, 55, _, 5
pop: _, _, 55, _, 5
annoy: _, 55, _, 5
pop: 55, _, 5

Then the next 'annoy' gets us in the loop again.

1189 is likely to be the next one never to appear, but I haven't thought

-----Jay Anderson
```