[seqfan] Re: A098550.

Vladimir Shevelev shevelev at bgu.ac.il
Fri Dec 12 18:00:03 CET 2014


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/
>> >
>>
>> _______________________________________________
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>
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