[seqfan] Re: A098550.

Tue Dec 9 16:16:06 CET 2014

In comment of A251417, Neil wrote "I would very much like to
understand the structure of this sequence."
Here I give a formula for A251417.
If A098550 is a permutation of the positive integers, then A098551 is well
defined. Let f(n)=A098551(A251595(n)). Then one can prove that
A251417(n) = f(n) - f(n-1), n>=2.
Example. Let n=12, then A251595(12)=19, while A251595(11)=18.
We have A098551(19)=43 and A098551(18)=31. So A251417(12)=
43-31=12.

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Reinhard Zumkeller [reinhard.zumkeller at gmail.com]
Sent: 08 December 2014 17:49
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A098550.

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/
> >
>
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