[seqfan] Re: A098550.

Mon Dec 8 21:34:05 CET 2014

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/
> > >
> > > _______________________________________________
> > >
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> > >
> >
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> >
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>
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