[seqfan] Re: A098550.

Vladimir Shevelev shevelev at bgu.ac.il
Sun Dec 14 21:28:31 CET 2014


Sorry, looking at my A249943, I gave wrong examples,
e.g., it should be 43 repeats 4 times,  43=a(19) and  4=23-19;
51 repeats 6 times, 51=a(29), and 6=29-23, etc.

Best regards,
Vladimir

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 14 December 2014 18:15
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A098550.

Two words on the discovery of my formula in A251621 connected with
A001223. Since I did my A249943 by handy (using table in A089550), it
stucked in my head that the number of the same terms often is close
to the distance between some nearest primes. For example, 43 repears 4 times
which is 47-43, 51 repears 6 times which is 59-53, 61 repeates 2 times (61-59),
62 repeats 6 times (67-61), 79 repeats 4 times (83-79), etc. I was sure that it
should be a somewhat connection. Further I noted a conjectural inequality
(see formula in A249943) and a little (for two days) digressed from lengths,
but tonight of 6.12 I have found a crossref in A249943 to A251621 on what that stucked in my head !
Morning I began to compare and compare the data with A001233;
firstly I did not see a direct connection and not hope on a full success, but soon I
found the PARALLEL consecutive coincidences!! For me it was a wonderful success!
I sent the formula to A251621 and to fan. list, but tonight Brad wrote in fan. list
that it is equivalent to his conjectures at A251416 which I saw for the first
time. Since I did not understand its former definition, I not understood
Brad's arguments. As I already wrote, later, after the change of the name of
A251416, I indeed deduced by induction my formula from Brad's conjectures.

Best regards,
Vladimir

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Brad Klee [bradklee at gmail.com]
Sent: 12 December 2014 21:13
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A098550.

Hi Vladimir,

I think it would be more appropriate to call that class "Shevlev's class".
I have been thinking about an infinite class, but it is slightly different.
I get an infinite number of sequences by leaving the sequence definition
the same, and by changing the axiom. This is described in the introduction
to the article I am writing about A098550 ( introduction included below ).

I think that the axiom of A098550, { A[1] = 1, A[2] = 2, A[3] = 3 }, is
somewhat arbitrary. It would be possible to ask the same sorts of question
and do the same sorts of analysis on any sequence obeying the defining
properties, but with axiom
{ A[1] = ?, A[2] = ?? } where ? and ?? are co-prime and not equal to one.
These sequences will never be a permutation of the integers because 1 will
not appear. So it could be better to consider { A[1] = 1, A[2] = ?, A[3] =
??}.

I think it will be possible to write arguments that will cover any such
sequence.

Thanks,

Brad

- - - -

I. Introduction

Consider a sequence defined recursively from some axiom {A[1],A[2],...}
with four definitive properties

d1. A[n] co-prime A[n-1] ,
d2. A[n] not co-prime A[n-2] ,
d3. No number occurs twice ,
d4. When values A[1] - A[n-1] are defined, A[n] is the minimum number
satisfying 1 & 2 & 3.

Any number of sequences can be generated by setting axiomatic values for
A[1] and A[2], which then uniquely determine the entire sequence according
to properties 1-4. Notice that the number 1 cannot be included in such
sequences. Arguments used throughout this article depend little on the
axiomatic initial values, and rely mainly on properties d1-d4. After a
first example, generalization will extend arguments to sequences of
arbitrary axiom.

The first example of a sequence satsifying properties d1-d4 for n > 3 is
OEIS sequence A098550. This sequence starts from three axiomatic values

A[1] := 1,
A[2] := 2,
A[3] := 3.

The first few terms of A[n] are

1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39,
11...

This part of the sequence already contains all numbers 1-12. A computer
capable of determining hundreds of thousands of terms can verify that the
sequence contains larger and large parts of the natural numbers when the
number of computed terms increases. Observing this trend, one might think
that the series eventually contains all natural numbers. Many of the
"obvious" properties of the sequence are difficult to deduce only from the
definitive properties d1-d4 and the axiom.

Subsequent sections of the article will reveal apparent properties of the
sequence, especially by the definition of derived sequences. A few
conjectures summarize empirical discoveries, and present mysteries beyond
that go beyond the provable facts. Avoiding the conjectures, this article (
draft editor: attempts ) to show that

Lemma 1. The sequence contains an infinite number of terms divisible by p_m
and arbitrary prime number.
Lemma 2. The sequence contains all prime numbers.
Lemma 3. The sequence always visits the smallest missing number.
Theorem 1. The sequence A098550 is a permutation of the natural numbers.

II. Related Sequences, Observations, Conjectures
        a. Spectral Decomposition

( draft editor: perhaps extend to include graph of A[n] )

Properties d1 and d2 are evaluated for a triplet {A[n], A[n-1], A[n-2]} by
listing and comparing the prime factors of each term A[n]. Better
understanding of the sequence should follow from patterns that show the
occurence of prime factors throughout the series. One such guiding pattern
occurs in the prime factor index table, defined as

T[m,i] := the index n in sequence A098550 for ith occurence of a number
divisible by the mth prime number.

For example, the first ten occurences of numbers divisible by primes 2, 3
... 29 determine a 10 x 10 segment of the table

2, 4, 6, 8, 10, 12, 14, 16, 18, 20
3, 5, 7, 10, 12, 17, 19, 21, 24, 26
7, 9, 11, 13, 16, 18, 26, 32, 34, 36
8, 13, 15, 17, 27, 38, 40, 42, 44, 58
20, 22, 24, 34, 46, 66, 68, 73, 91, 100
21, 23, 25, 36, 38, 54, 83, 95, 108, 120
28, 30, 32, 35, 67, 70, 111, 115, 126, 141
41, 43, 45, 47, 72, 82, 113, 123, 163, 166
49, 51, 53, 56, 74, 93, 149, 157, 197, 228
59, 61, 63, 96, 105, 121, 188, 198, 250, 259

Plotting the quantity T[m,i] / i ( figure 1. ) for various values of m
gives a graph with a spectral appearance in the limit of increasing i; that
is, after some initial disorder, each series settles to an approximately
constant value located around p_m, the mth prime number. This leads to the
first "obvious" conjecture

Conjecture 1. For each m there exists some real \lambda_m such that Limit (
k --> infinity ) 1/k Sum_{i=1}^{k} T[m,i] / i = \lambda_m, and the limit
exists.

A proof of this conjecture would give excellent understanding of the
sequence's behavior, but knowledge at that level of detail is not strictly
necessary for understanding some basic properties of the sequence. Lemma 1
provides a weaker fact that is nevertheless sufficient for reaching lemmas
2 & 3 and finally theorem 1. To understand that the sequence contains all
primes, it is only necessary to know that the table T[i,j] has a well
defined value for all i, j in N, the natural numbers.

On Fri, Dec 12, 2014 at 11:00 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
wrote:
>
> A generalization. Consider the set of permutations {A} of the positive
> integers
> and construct for every A the sequences K=K(A), S=S(A) and Z=Z(A)
> corresponding to A251416, A249943 and A251621 respectively. If, for a given
> permutation A,  the Klee's conditions (conjectures) for primes hold for the
> sequence K (beginning with a place), then we say that the permutation A is
> in
> Klee's class. Then we have a formula: for n>=c_1,  (Z(A))(n) =
> A001223(n-c_2),
> where c_i =c_i(A) are positive constants.
> Note that the identity permutation A: N->N, evidently, is not in Klee's
> class,
> since in this case the sequence K does not exist (is empty).
>
> Best regards,
> Vladimir
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir
> Shevelev [shevelev at exchange.bgu.ac.il]
> Sent: 11 December 2014 09:00
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A098550.
>
> Finally, we have an explicit conjectural formula for prime(n), n>=8,
> based on
> A098550 and simple operations (A098550->A249943 -> A251621) :
> prime(n) = 19 + sum{i=9,...,n}A251621(i+4), n=8,9,...
> (as always, sum{a,...,b}=0, if b<a).
>
> Best regards,
> Vladimir
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir
> Shevelev [shevelev at exchange.bgu.ac.il]
> Sent: 10 December 2014 20:03
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A098550.
>
> Dear Brad,
>
> After the best Neil's restatement of the definition of
> A251416, I see that I did not right understood former
> definition, sorry for my English (I am still poorly understand
> an equivalence of the name and the comment being
> the name two days ago, because of the difficult for me
> structure of this phrase).  Now I see that, by a simple
> induction, my conjecture follows from yours. I hope
> that the inverse statement is also true.
>
> Best regards,
> Vladimir
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Neil Sloane [
> njasloane at gmail.com]
> Sent: 09 December 2014 21:00
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A098550.
>
> Vladimir, your formula is just a restatement of the definition of
> A251417.  I should have said that it would be nice to
> have a formula for this sequence that was not based on A098550!
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Tue, Dec 9, 2014 at 10:16 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
> wrote:
> >
> > 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/
> >> >
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> > _______________________________________________
> >
> > 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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