[seqfan] Re: Very bounded sequences

[Tomasz Ordowski]

P.S. Thanks to Neil Sloane, I have submitted the sequence
https://oeis.org/history/view?seq=A307486&v=19
with a comment by Carl Pomerance.

śr., 10 kwi 2019 o 16:00 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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/
>> >
>> >
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>