[seqfan] Re: A098550.

Sun Dec 7 21:23:38 CET 2014

Hi All,

Still working on the permutation proof, and the latest draft seems pretty
good.

I have found some more interesting sequences, that really help to
understand the sequence.

Define the table T[ i , j ] where element the element t_ij gives the index
in A098550 of the j_th occurrence of a multiple of prime p_i .

with seq some length of A098550 a portion of the table is given by

t = Position[FactorInteger[#][[All, 1]] & /@ seq, Prime[#] ][[All,
      1]] & /@ Range[15];

Plots by row show partitioning similar to partitioning observed in the
graph of sequence values. The rows are approximately linear sequences.

ListPlot[ t ]

ListPlot[ Map[#/Range[Length@#] &, t[[1 ;; 8]] ] ,
 PlotRange -> {0, 20}]

This graph and some counting logic have convinced me that it is possible to
prove this table is infinite in both directions, which in turn implies that
no B-loops or D-loops can occur, which in turn implies that A098550 is a
permutation of natural numbers.

The idea is to bound the search for the next multiple of p_m, and show that
the next multiple of p_m must occur within that bound. From reading the
graphs, it looks like a quadratic bounding function would work, but I have
used a power-law in my analysis.

Thanks,

Brad

 - - - -

Small block of 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

On Sun, Dec 7, 2014 at 11:28 AM, Brad Klee <bradklee at gmail.com> wrote:

> 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/
>>
>
>