# [seqfan] Re: New sequence

Neil Fernandez primeness at borve.org
Thu Apr 6 14:22:03 CEST 2017

```Hi,

Graeme McRae wrote:

> I also looked for even numbers such that *all* the values E+pn are
> prime. The only such even numbers I found smaller than 1000 were 8 and
> 12. In fact, the only even numbers smaller than 1000 such that all but
> one of the values E+pn are prime are 8, 12, 18, 24, and 30.

The only prime pair summing to 8 is {5,3}, and 8 + {5,3} are both prime.
-"- summing to 12 is {5,7}, and 12 + {5,7} are both prime.

10 is not on the list if we count the summands with multiplicity
({3,5,5,7}) but it is if we don't. The same is true for 6.

I get

(6, 10,) 18, 24, 30

for E < 20000.

Another nice sequence is

a(n):= the number of pairs of primes {p_1, p_2} that sum to 2n and for
which 2n+p_1 and 2n+p_2 are both prime.

For this I get

0,0,0,2,2,2,1,2,3,2,1,5,1,0,1,0,5,1,1,5,0,1,6,...

And the numbers for which a(n/2) is a maximum run

8, 18, 24, 42, 60, 90, 120, 210, 330, 420, 630, 840, 1050, 1260, 1470,
1680, 1890

Neil

--
Neil Fernandez

```