[seqfan] Re: A098550.
apovolot at gmail.com
apovolot at gmail.com
Thu Dec 11 14:33:05 CET 2014
Perhaps then you could also relate A251621 with A153143 in some kind of formula of the same sort ...
Alexander R. Povolotsky
> On Dec 11, 2014, at 2:00 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
>
> 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
>>>>
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>>>>
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