Generalization of Brun's Constant

[Zeta-Flux]

Theoretically, will B(3) be the maximum of the B(n)?

[R. Gerbicz]

Originally Posted by R.D. Silverman

You should expect to see increases in the value of B(n) at n = 30, 210, ....

I don't think because yes you have got more primes for those sequences but in the beginning of the sequences you have got only one small prime, because the other prime q=p+2*n>2*n is too large so 1/q=1/(p+2*n)<1/(2*n) is too small.
Conjecture: B(n)>=2 only for the following n values: n=3,6,9,12,15,18,21,30,33,45,60,105,210
The approximations for these B(n):

Using primes up to x=2^27
B(3)=2.442231269392689804114732068
B(6)=2.201030111019210864675522552
B(9)=2.003385568331636746610271054
B(12)=2.124422255535451994857424058
B(15)=2.388569586152179981872294340
B(18)=2.030799435427551083938044648
B(21)=2.102082829864169356075057696
B(30)=2.204198213726053404129335028
B(33)=2.053021584235230176112771307
B(45)=2.152161728195823869080966493
B(60)=2.067666234021665015724540739
B(105)=2.119764793288551158021767410
B(210)=2.039955879804044710971032031

Originally Posted by Zeta-Flux

Theoretically, will B(3) be the maximum of the B(n)?

From this Zeta-Flux I think that B(3) will be the maximum.

[Jens K Andersen]

Originally Posted by R.D. Silverman

Another interesting question is exactly where p first occurs.
For example one might think that B_{3*5*7} = 11, but B_105 = 13
because 11+210 = 221 = 13*17. Does 13 occur before 11?
Where does 11 first appear?

Does q appear before p infinitely often when q > p?

If someone is interested, they might want to write some code to
determine the first time that a given prime p appears.

I did that for 2n < 10^12 in 2004: 2n < 3*10^10 , 3*10^10 < 2n < 10^12, and B_n champions

The largest found B_n was B_(496562420542/2) = 3307.
I submitted some related sequences to OEIS: First occurrence of nth prime

[R.D. Silverman]

Originally Posted by Jens K Andersen

I did that for 2n < 10^12 in 2004: 2n < 3*10^10 , 3*10^10 < 2n < 10^12, and B_n champions

The largest found B_n was B_(496562420542/2) = 3307.
I submitted some related sequences to OEIS: First occurrence of nth prime

I was not aware of these references. Thank you.