[seqfan] "nearest" twin primes

[Vladimir Shevelev]

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