# [seqfan] Re: Two branches in scatter graph of A102605

zak seidov zakseidov at yahoo.com
Fri Oct 27 19:49:20 CEST 2017

```Great! I did it for mod 10
and found nothing:all final digits 1,3,5,7,9 in n occur exactly 4 times each(from 20 combinations of final digits of primes p, q, r).
Also Alois Heinz (pers. comm.) made some attemptsnot specifying them.BTW Actually each "branch" has a finer structurewith almost 5 subbranches - like Saturn rings.
Zak

From: M. F. Hasler <seqfan at hasler.fr>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Friday, October 27, 2017 7:23 PM
Subject: [seqfan] Re: Two branches in scatter graph of A102605

>
> On Fri, Oct 27, 2017 at 3:58 AM, Neil Sloane <njasloane at gmail.com> wrote:
> > Here is a piece of the b-file:
> > ...
> > 123 125
> > 124 91
> > 125 135
> > 126 128
> > 127 89
> > 128 143
> > 129 127
> > 130 94
> > 131 146
> > 132 127
> > 133 102
> > 134 147
> > 135 141
> > 136 105
>

It appears that for n = 1 (mod 3) there are about 30% less ways of writing
2n+1 as sum of 3 distinct primes.
If n=3k+1, then 2n+1 = 2(3k+1) +1 = 6k+3.

If you consider all possible combinations of 3 primes which can each be
either 1 or 5 (mod 6),
then only in 2 cases the sum will be 3 (mod 6), and in 3 cases it will be 1
(mod 6), and in 3 cases it will be 5 (mod 6).

This might at least partially explain the bias of -30% for n=1 (mod 3)
w.r.t. the other cases n=0 and n=2 (mod 3).

--
Maximilian

> On Thu, Oct 26, 2017 at 6:51 PM, zak seidov  wrote:
> > > Any explanation of two "branches" in  https://oeis.org/A102605/graph
>
> > A102605: Number of ways of writing 2n+1 as p+q+r where p,q,r are primes
> with p < q < r.
>

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