[seqfan] A conjecture in A284919

Bob Selcoe rselcoe at entouchonline.net
Tue Apr 11 00:17:20 CEST 2017

Hi Seqfans,

A284919 is Claudio Meller's interesting new sequence: "Even integers E such 
that there is no prime p < E with E - p and E + p both prime".

It appears that except for A284919(4) = 6, all terms in the sequence are 
coprime to 3 (Thanks Robert Israel, for checking this to n = 5*10^6).  [Note 
that as of this writing, the published version doesn't show it as a 
conjecture, but it's under review and hopefully will be corrected soon, if 
it hasn't already].

This is equivalent to the conjecture:  For n = 6*k (k > 1), there must be a 
prime p < 6k such that when 6k-p is prime, 6k+p also is prime.

This doesn't seem easy to prove (or disprove).  Does anyone have any ideas 
about this?

Also, assuming the conjecture is true, might it suggest that at some point, 
all terms might also become coprime to some numbers > 3?

Bob Selcoe

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