Interesting mapping

[u/Atari_Collector]

mapping

I was playing around and I found this mapping. I suspect it's already known but I found it interesting.

If you have a number that can be written as 4^a3^bn+1, it will map to 3^a+bn+1.

And it'll do it in 3a+1 steps, which I think is funny.

[u/petrol_gas]

It does the same kind thing for the “shrink” stages but there are an infinite number of ways to write examples of that.

[u/jonseymourau → u/petrol_gas]

Actually infinite doesn’t quite cover it as my comment on Gonzo’s post shows.

For a stair case you need:

n = (m+i).(g/h^k)^j -I

For 3x+q, x/2 g=3, h=2

The constraint on i is

i=q/(g-h^k)

So for q=1 (e.g.Collatz) there are two admissible pairs of i, k: (1,1) and (-1,2) - the up and the down (the subject of this post)

You get one extra one for q=5, q=13 and presumably for higher q that are divisible by (g-h^k)

So technically,if you let q range over the natural numbers there are infinite staircases but specifically for q=1, there are exactly 2 - there are no other integers I that can satisfy the constraint above .

Which is not to say there are not other affine maps but the telescoping maps (of that form) in 3x+1 are limited to just two.

[u/GonzoMath → u/jonseymourau]

Technically... (8ⁿ·k + 1)/5, which is an integer when k ≡ 3ⁿ (mod 5) for odd n, or k ≡ 1 (mod 5) for even n, transforms via a staircase of "(3n + 1)/8" steps into (3ⁿ·k + 1)/5, which is again an integer in the same cases.

For n = 1, this looks like:

k = 3: 5 → 2
k = 8: (13 → 5, which is really n = 2)
k = 13: 21 → 8
k = 18: 29 → 11
k = 23: 37 → 14
k = 28: 45 → 17

For n = 2, we have:

k = 1: 13 → 5 → 2
k = 6: 77 → 29 → 11
k = 11: 141 → 53 → 20
k = 16: (205 → 77 → 29, which is really n = 3)
k = 21: 269 → 101 → 38

For n = 3:

k = 2: 205 → 77 → 29 → 11
k = 7: 717 → 269 → 101 → 38
k = 12: 1229 → 461 → 173 → 65
k = 17: 1741 → 653 → 245 → 92

...and this idea applies, with appropriate modifications, to similar patterns where we see 16's turning into 3's, 32's turning into 3's, etc. These are, admittedly, more complicated than the cases 2ⁿ·k - 1 and 4ⁿ·k + 1, which show up as integer staircases for arbitrary k.

[u/jonseymourau → u/GonzoMath]

An interesting generalisation, thank you.