[seqfan] Re: Very bounded sequences

[Tomasz Ordowski]

Robert,

thank you for your interest in the topic. See PS.

Neil Sloane asked me to submit this sequence to OEIS,
but unfortunately I do not have free editing right now.

Thomas

P.S. I have something new:

a(0) = 1, for n > 0, a(n) is the smallest prime p such that
m = a(0)+a(1)+...+a(n-1)+p is composite and p does not divide m.

a(n) : 1, 3, 5, 5, 11, 2, ... (extensive data below).

Is this sequence bounded? Probably not!

Sincerely,

Thomas
______________
a(n), smallest n :
============
3, 1
5, 2
11, 4
13, 23
17, 1115
19, 23632
23, 67635
29, 26106256
31, 61953072
37, 363243369
...
[Amiram Eldar]

śr., 10 kwi 2019 o 14:12 <israel at math.ubc.ca> napisał(a):

> P(n) = a(0)...a(n-1) grows exponentially, so heuristically the probability
> that 1+2*P(n-1) and 1+3*P(n-1) are both prime (resulting in a(n+1)>=4) is
> on the order of n^(-2). Since sum_n n^(-2) converges, we should expect
> this
> to occur only finitely many times. Of course this is not a proof. But if
> the reason your conjecture is true is "only statistical", maybe we should
> not expect a proof to be possible.
>
> Cheers,
> Robert
>
> On Apr 10 2019, Tomasz Ordowski wrote:
>
> >Dear SeqFans,
> >
> >I defined an interesting 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 that are natural
> >numbers.
> >
> >Conjecture: For any initial term a(0) > 0, a(n) > 3 only for finitely many
> >n >= 0.
> >
> >The question is how to prove that all these sequences are bounded, so
> >bounded?
> >
> >It seems that a(0) = 21 is the smallest initial term such that a(n) = 2 or
> >3 for every n > 0.
> >
> >Note that if a(0) is a Sierpinski number, then a(n) = 2 for every n > 0.
> >
> >How to explain the occurrence of only twos and threes in the tails?
> >
> >Are similar sequences described in the literature?
> >
> >Best regards,
> >
> >Thomas Ordowski
> >
> >--
> >Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
>
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