[seqfan] Re: "nearest" twin primes

[Vladimir Shevelev]

Dear SeqFans,

It is interesting a continuation. 
Consider a new sequence:
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;
a(n) is the smallest m such that
B_(p+2)(m)-B_p(m)>6 or a(n)=0 
if there is no such m.
Peter found (n>=2)
0, 13, 0, 0, 0, 9, 0, 11, 11, 5, 3, 15, 3, 7, 3, 0, 3, 0, 3, 5, 7, 3, 11, 5, 3, 5, 11, 3, 9, 3, 3, 7, 3, 5, 5, 3, 5, 3, 5, 11, 3, 5, 0, 0, 5, 5, 7, 5, 13, 7, 0, 5, 3, 3, 3, 3, 7, 3, 3, 3, 5, 3, 7, 3, 3, 0, 3, 5, 5, 3, 11, 11, 5, 3, 5, 7, 5, 3, 0, 3, 3, 3, 3, 3
I conjecture that a(n)<=15 (or 16 or 17).
Peter found that a(470)=17  and up to 200000 
not found any term more than 17. Moreover,
he found only the terms 0,3,5,7,9,11,13, 15 and 17.
Now I am pleased to inform that I proved this.
In the nearest days I hope to publish with Peter
a paper in arxiv.
Note that I cannot to submit the above sequence
since I already have the maximal possible 3
submitted but yet unpublished sequences.

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 22 September 2016 17:10
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: "nearest" twin primes

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,
Vladimir

________________________________________
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,
Vladimir

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