[seqfan] Re: A098550.

[Vladimir Shevelev]

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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