# [seqfan] b-Zahlen

David Sycamore djsycamore at yahoo.co.uk
Fri Dec 7 18:10:56 CET 2018

```Consider: « Even number 2n >= 4 such that 2n-p is prime, where p is the greatest prime <= 2n -1 »

This generates what at first looks like A005843 (the even numbers), except for 0, 2. This holds until n = 49 (2n = 98) because then p = 89 and  98 - 89 = 9, which is composite. Continuing, we find more numbers which don’t fit the initial pattern: 98,122,128,148,190...
This turns out to be A244207, the so called « b-Zahlen », apparently first noted by Nils Johan Pipping (1890 - 1982).

Let b_k denote the even numbers 2n such that the greatest prime q < 2n-1 having the property that 2n-q is prime is the k-th prime below (less than) 2n-1. Then A244207 is the sequence of even numbers 2n such that k > 1.

And we have the « b_k-Zahlen » sequences:

b_1 : A005843 less {0,2} and all numbers in A244207.

And then the following subsequences  of A244207 :

b_2 : 122,128,148,190,208,220,250,
292,302,326,332,410,418,430......

b_3 : 98, 346, 368, 398, 458, 518, 586, 640, 692.....

b_4 : 854, 908.....

b_5 : 308, 488, 556....

This is as far as I’ve got;  nothing yet for k >= 6.

Also: a(n) = smallest number in sequence b_n :

4, 122, 98, 854, 308...

None of the above are in Oeis, although the b_4 numbers seem to appear in A244408, A279040 and A038433.

I doubt  very much if any of this is of any use, but is it of interest to Oeis ?

Regards

David.

```