[seqfan] Re: Sierpinski sequences

[Tomasz Ordowski]

P.S. SUPPLEMENT.

It seems that a(0) = 21 is the smallest initial term such that a(n) = 2 or
3 for every n > 0.

The number 21 is called "pip". Find further probable "pip numbers".

Recall the recursive definition of my sequences:

For n > 0, a(n) is the smallest k > 1 such that a(0)*a(1)*...*a(n-1)*k + 1
is composite.

I gave them the name "Sierpinski sequences".

Can also consider the "Riesel sequences", defined:

a(n) is the smallest k > 1 such that a(0)*a(1)*...*a(n-1)*k - 1 is
composite.

Note that if a(0) is a Riesel number, then a(n) = 2 for every n > 0.

Simultaneously the "Sierpinski and Riesel sequences":

a(n) is the smallest k > 1 such that both a(0)*a(1)*...*a(n-1)*k +- 1 are
composite.

Finally, the "Sierpinski or Riesel sequences": at least one of the ... +- 1
is composite.

Thank you for your attention and I ask for comments.

Thomas Ordowski
____________________________________
https://en.wikipedia.org/wiki/Riesel_number

czw., 4 kwi 2019 o 18:53 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear SeqFans!
>
> Here is an example of the "Sierpinski sequence":
>
> a(0) = 3; a(n) = smallest k > 1 such that 1 + a(0)*a(1)*...*a(n-1)*k is
> composite.
>
> 3, 3, 3, 2, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2,
> 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, ...
>
> a(n) = 4 for n = 4 and 39,
> a(n) = 3 for n = 0, 1, 2, 5, 6, 7, 14, 20, 25, 56, 90, 119, 316, 330,
> 1268, 1604, 1805, 1880, 1984, 2950, 3386, 3712, 4532, 4874, 8968, 18178,
> 19454, ...
> a(n) = 2 for others n < 20000.
> Data from Amiram Eldar.
>
> Similar tails have sequences with other initial terms being natural
> numbers.
>
> Conjecture: For any initial term a(0) > 0, only finitely many terms a(n) >
> 3.
>
> The question is how to prove that these sequences are so bounded?
>
> Note that if a(0) is a Sierpinski number, then a(n) = 2 for every n > 0.
> Hence the name of my sequences, in honor of this great mathematician.
>
> Best regards,
>
> Thomas Ordowski
> _______________________________________
> https://en.wikipedia.org/wiki/Sierpinski_number
>
>