[seqfan] Re: A098550.

Mon Dec 8 19:34:21 CET 2014

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
> > >
> > > _______________________________________________
> > >
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> > >
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