[seqfan] Re: A098550.

[Brad Klee]

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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