# [seqfan] Re: a permutation of the naturals

Andrew Weimholt andrew.weimholt at gmail.com
Wed Oct 28 22:23:21 CET 2009

```Yes, I agree. Until someone can prove that no number is catapulted an
infinite number of times,
I should probably refer to the sequence as a "conjectured permutation
of the naturals (plus zero)".
I also agree that 0 catapults 1, giving b(1)=1

The total offset by which n is catapulted is interesting, and can be
quite larger than the offset to its final position, due to numerous
leftward displacements when other numbers are catapulted over it.

Andrew

2009/10/28 Benoît Jubin <benoit.jubin at gmail.com>:
> On Wed, Oct 28, 2009 at 7:50 PM, Andrew Weimholt
> <andrew.weimholt at gmail.com> wrote:
>> I will submit the sequences tonight, along with one more related sequence.
>> Each number (assuming there is no number catapulted an infinite number
>> of times)
>
> Actually, it would be nice to prove this assumption since we don't
> know yet if your sequence is a permutation of the naturals, and hence
> if the "inverse permutation" is everywhere defined.  Anyone can come
> with a proof?  I tried a little bit, in vain.
>
> Also, the sequence "number of times catapulted" would change to
> b(0)=0, b(1)=1, b(2)=3, ... (because 1 is catapulted one time by 0
> position)
>
> and another sequence worth submitting would be "total offset by which
> n is catapulted": c(0)=0, c(1)=0, c(2)=1+3+4=8, ...
>
> Benoit
>
>
>  will eventually catapult one and only one other number, so
>> we can define another sequence in which a(n) is the number that n
>> catapults in the generation of the first sequence.
>>
>> I will also incorporate Benoit's suggestion of beginning with 0, which
>> does not affect the
>> rest of the sequence.
>>
>> One interesting observation, when collecting more terms: 29 is
>> catapulted 17 times and ends up in a position offset by more than
>> 150000 from its original position.
>> Also, 0,1,54 are the only numbers I found so far with a(n)=n.
>>
>> Andrew
>>
>> On 10/28/09, Benoît Jubin <benoit.jubin at gmail.com> wrote:
>>> Interesting sequence, I'm looking forward to seeing its graph.
>>>
>>>  I suggest you begin the sequence at 0 since it does not require any
>>>  special treatment.
>>>
>>>  Benoit
>>>
>>>
>>>
>>>  On Wed, Oct 28, 2009 at 10:10 AM, Andrew Weimholt
>>>  <andrew.weimholt at gmail.com> wrote:
>>>  > This idea was inspired by one of Eric Angelini's recent posts (angry numbers).
>>>  >
>>>  > We start with the natural numbers in their normal positions,
>>>  > and then the number in position 1 (which happens to be 1), catapults
>>>  > the number to its right to a position 1 further to the right.
>>>  > So after the first step, we have 1,3,2,4,5,6,7,8...
>>>  > Then the number now in position 2, (which is 3), catapults the number
>>>  > to its right (which is 2) to a position 3 further to the right
>>>  > Now we have, 1,3,4,5,6,2,7,8...
>>>  > In the nth step, the number now in the nth position (which will be
>>>  > a(n)) catapults the number to its right to a position a(n) further to
>>>  > the right.
>>>  >
>>>  > The sequence, beginning at n=1 is...
>>>  >
>>>  > 1, 3, 4, 6, 7, 5, 10, 2, 13, 12, 14, 16, 18, 19, 21,
>>>  > 23, 8, 25, 15, 28, 17, 24, 32, 33, 20, 36, 22, 38, 40, 41, 42,
>>>  > 44, 45, 47, 31, 35, 50, 52, 27, 55, 11, 58, 59, 61, 63, 64, 66,
>>>  >
>>>  > The inverse permutation is...
>>>  >
>>>  > 1, 8, 2, 3, 6, 4, 5, 17, 152, 7, 41, 10, 9, 11, 19,
>>>  > 12, 21, 13, 14, 25, 15, 27, 16, 22, 18, 57, 39, 20,
>>>  >
>>>  > The following sequence gives the number of times n is catapulted
>>>  >
>>>  > 0, 3, 0, 0, 1, 0, 0, 2, 6, 0, 3, 1, 0, 0, 1,
>>>  > 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0,
>>>  >
>>>  > Not sure these are worth submitting, but thought I'd at least share
>>>  > them the seqfan list
>>>  >
>>>  > Andrew
>>>  >
>>>  >
>>>
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