[seqfan] Re: A098550.

Brad Klee bradklee at gmail.com
Mon Dec 8 22:05:56 CET 2014 

Hi,

If you leave a(0) = 0, then I think you will need to change the the
plaintext definition.

" a(0) = 0; for n>0, a(n) is the smallest integer not already in the list
with a composite common factor with a(n-1). "

1 > 0, but a( 1 - 1 ) = a( 0 ) = 0 does not have composite common factor
with any integer ? ! ? !

But if you read the computer definition it says something else. I find it
more confusing including 0, but can't speak for everyone.

Thanks,

Brad

On Mon, Dec 8, 2014 at 2:34 PM, Frank Adams-Watters <franktaw at netscape.net>
wrote:

> We have a suggestion here that the a(0) term should be omitted, leaving
> offset 1. L. Edson Jeffery in a pink-box comment suggests changing the
> offset to 1 without taking 0 out of the sequence. In my opinion, these two
> arguments cancel out in favor of leaving the sequence as it is. The
> sequence starting from a(1) is (presumably) the composite numbers
> reordered; having that start from a(2) seems wrong to me. On the other
> hand, a unique plausible a(0) does exist, so the usual policy is that that
> should be included in the sequence: someone might make a search including
> that term.
>
> I would be okay with changing "every composite integer" to "every
> composite number". The comment does not imply that a(0) is composite.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Brad Klee <bradklee at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Mon, Dec 8, 2014 1:00 pm
> Subject: [seqfan] Re: A098550.
>
>
> Hi Reinhard,
>
> These conjectures are more surprising and questionable to me.
>
> But I am a bit confused about how https://oeis.org/A251756 has been
> defined.
>
> I think we should consider redefining A251756.
>
> The comment is not strictly correct because it isn't a permutation of
> composites in Z ( for Zahlen ) because Z includes negative numbers. Neither
> is it a permutation of composite N ( for natural ) because N does not
> include zero according to OEIS.
>
> Considering the comment 1, It's strange to have zero in there at all
> because zero isn't in https://oeis.org/A002808. The definition uses 4 as
> the axiom and prepends zero without function.
>
> My suggestion is to redefine this series to eliminate a(0) = 0 and start
> from a(1) = 4. And then rewrite the comment to say:
>
> "It appears the sequence contains every natural number."
>
> This does not affect your conjectures, except for simplifying 2.
>
> Thanks,
>
> Brad
>
>
>
>
> On Mon, Dec 8, 2014 at 9:49 AM, Reinhard Zumkeller <
> reinhard.zumkeller at gmail.com> wrote:
>
>  1. conjecture: A098550 is a permutation of the positive integers
>> 2. conjecture: A251756 is a permutation of the composites (except
>>
> initial
>
>> 0)
>> 3. conjecture: the preceding two conjectures are equivalent.
>>
>> 2014-12-08 9:51 GMT+01:00 Vladimir Shevelev <shevelev at bgu.ac.il>:
>>
>> > Dear Brad,
>> >
>> > I back to understanding of what you did conjecture in A251416
>> > with respect to A098550:
>> >
>> > 1) Every prime is a term of A098550;
>> > 2) All primes in A098550 follow in the natural order;
>> > 3) After n=c (c is a small constant) every prime is maximum +1 among
>> > the consecutive positive integers which at the moment already are
>>
> terms
>
>> > of A098550;
>> > 4) After n=c, every prime p first is necesarily missing as the
>>
> following
>
>> > integer
>> > after the consecutive positive integers from 3), i.e., passes
>>
> infront of
>
>> it
>> > at least p+1.
>> >
>> > If so, then I understand that our conjectures are equivalent.
>> >
>> > Thanks,
>> >
>> > Vladimir
>> >
>> >
>> > ________________________________________
>> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir
>> > Shevelev [shevelev at exchange.bgu.ac.il]
>> > Sent: 07 December 2014 22:26
>> > To: Sequence Fanatics Discussion list
>> > Subject: [seqfan] Re: A098550.
>> >
>> > I did not understand your proof of
>> > the equivalence. It seems me to be not sufficient.
>> >
>> > Thanks,
>> >
>> > Vladimir
>> >
>> > ________________________________________
>> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Brad Klee
>>
> [
>
>> > bradklee at gmail.com]
>> > Sent: 07 December 2014 19:28
>> > To: Sequence Fanatics Discussion list
>> > Subject: [seqfan] Re: A098550.
>> >
>> > Your conjecture is fully equivalent to the conjecture I made for
>>
> A251416.
>
>> >
>> >  https://oeis.org/A251416
>> >
>> > Assume my conjecture true ( false ), the sequence always changes (
>> > sometimes doesn't change )  minimum value to the next prime. Then a
>>
> prime
>
>> > and all terms up to next prime have ( don't have ) the same value
>>
> for
>
>> > A249943. The number of those terms is ( is not ) the difference
>>
> between
>
>> > primes, so your conjecture must also be ( true ) false.
>> >
>> > The reverse case should be similar.
>> >
>> > Thanks,
>> >
>> > Brad
>> >
>> >
>> >
>> > On Sun, Dec 7, 2014 at 4:14 AM, Vladimir Shevelev
>>
> <shevelev at bgu.ac.il>
>
>> > wrote:
>> >
>> > > Sequences A249943 and A251621 are directly connected with A098550.
>> > > On the other hand, we conjecture that A251621 is directly
>>
> connected
>
>> > > with prime gaps (A001223). Namely, for n>= 13, we have A251621(n)
>> > >  = A001223(n-5).
>> > >
>> > > Best regards,
>> > > Vladimir
>> > >
>> > > ________________________________________
>> > > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of L. Edson
>> > > Jeffery [lejeffery2 at gmail.com]
>> > > Sent: 04 December 2014 08:52
>> > > To: seqfan at list.seqfan.eu
>> > > Subject: [seqfan] Re: A098550.
>> > >
>> > > Since there has been so much discussion about A098550, I wanted to
>> > mention
>> > > that for the related sequence A098548, the sequence A of first
>> > differences
>> > > is
>> > >
>> > > A = {1, 1, 1, 5, 1, 11, 1, 5, 1, 5, 1, 5, 1, 11, 1, 5, ...}.
>> > >
>> > > This sequence reminds me of Eric Rowland's A132199. However, here
>> > > composites definitely are present but appear to be quite sparse. I
>> > computed
>> > > the sequence for n < 10^6 and found only thirty-five composite
>>
> terms.
>
>> > Their
>> > > indices in A are the sequence
>> > >
>> > > B = {496, 8270, 16046, 23818, 31594, 39368, 47142, 54914,
>> > >      62688, 70460, 78236, 86010, 93782, 101556, 109332,
>> > >      117106, 124882, 126670, 132654, 140428, 148204, 155976,
>> > >      163752, 171526, 179300, 187076, 194850, 202618, 210394,
>> > >      218168, 225940, 233714, 241490, 249264, 257038}.
>> > >
>> > > From B, we have that A(126670) = 55, but it turns out that all of
>>
> the
>
>> > rest
>> > > of the indices k in B are such that A(k) = 27. I find that to be
>>
> rather
>
>> > > strange: there are of course composites in A, but why does 27 play
>> such a
>> > > prominent role among them (if in fact it does)?
>> > >
>> > > The distinct terms of {A(n)} (n < 10^6), arranged in increasing
>>
> order,
>
>> > are
>> > >
>> > > C = {1, 5, 11, 13, 17, 23, 27, 29, 37, 41, 55}.
>> > >
>> > > I did not try to find the index of the first occurrence of each
>>
> term
>
>> of C
>> > > in A. I already checked, and C starts off the same as A104110 but
>>
> is
>
>> not
>> > > the same sequence.
>> > >
>> > > Assuming that A(1000000) is not composite, then the number of
>>
> composite
>
>> > > terms in {A(n)}, for n <= 10^k, where (so far) k=0..6, is the
>>
> sequence
>
>> > >
>> > > D = {0,0,0,0,1,4,35}.
>> > >
>> > > I used the following Mathematica program for A:
>> > >
>> > > (* sequence A: *)
>> > > max := 10^3;
>> > > a := {1, 2, 3};
>> > > For[n = 4, n <= max, n++,
>> > >   If[GCD[n, a[[-1]]] == 1 && GCD[n, a[[-2]]] > 1,
>> > >     AppendTo[a, n]]];
>> > > Differences[a]
>> > > (* putting max = 10^6 took a long time to get the sequence *)
>> > >
>> > >
>> > > I know the base for D is somewhat arbitrary, but can anyone
>>
> extend any
>
>> of
>> > > B, C or D?
>> > >
>> > > Finally, if any of this is interesting enough to add to the
>>
> database,
>
>> > then
>> > > please go ahead and do it, as before.
>> > >
>> > > Ed Jeffery
>> > >
>> > > _______________________________________________
>> > >
>> > > Seqfan Mailing list - http://list.seqfan.eu/
>> > >
>> > > _______________________________________________
>> > >
>> > > Seqfan Mailing list - http://list.seqfan.eu/
>> > >
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
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>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
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>> >
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>> >
>>
>> _______________________________________________
>>
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