# [seqfan] Re: "nearest" twin primes

Thu Sep 22 16:10:31 CEST 2016

```Dear SeqFans,

I understood why the phenomenon happens.

Distinguish 3 cases:
a) Lesser of twin primes  p=6n-1=10k+1;
b) Lesser of twin primes  p=6n-1=10k+7;
c) Lesser of twin primes  p=6n-1=10k+9.
In case a) all differences B_(p+2)(n)-B_p(n)<=6
iff for some t>=0, we have 7 primes of the form
30t+11(=p), 30t+13, 30t+17, 30t+19, 30t+23,
30t+29,30t+31. In this case B_p merges with
B_(p+2) in n=17.
In case b) all differences B_(p+2)(n)-B_p(n)<=6
iff for some t>=0, we have only 5 primes of the form
30t+17(=p), 30t+19, 30t+23, 30t+29, 30t+31.
In this case B_p merges with B_(p+2) in n=11.
Finally in case c) all differences B_(p+2)(n)-B_p(n)
<=6 iff for some t>=0, we have also 5 primes of the
form 30t+29(=p), 30t+31, 30t+37, 30t+41, 30t+43.
In this case also B_p merges with B_(p+2) in n=11.
Therefore,  cases b) and c) occur much more often
than a).

Best regards,

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 21 September 2016 13:13
To: seqfan at list.seqfan.eu
Subject: [seqfan] "nearest" twin primes

Dear Seqfans,

I submitted the sequence A276848:

"For a lesser p of twin primes, let
B_(p+2) and B_p be sequences
defined as A159559, but with initial
terms p+2 and p respectively.
The sequence lists p for which
all differences B_(p+2)(n)-B_p(n)<=6."

In some sense a(n), a(n)+2 are "nearest"
twin primes (cf. A276826).

The first terms of the sequence
3, 11, 17, 29, 59, 227, 269, 1277, 1289,
1607, 2129,...
I proved that for all n>=2
B_(p+2)(n) - B_p(n)<6 (=4)
if and only if p=3.

On my opinion, it is very astonishing
that, although terms a(n)== 7 or 9 (mod 10)
occur often, the first terms a(n)==1 (mod 10)
are 11,165701,...
How to explain this phenomenon?

Best regards,