[seqfan] Re: No "-1" value in A047949?

Rick Shepherd rlshepherd2 at gmail.com
Sun Mar 14 19:29:14 CET 2010

At least A047949(n) >= A047160(n) > 0 for an infinite number of (prime) n.
This is a consequence of van der Corput's proof (1939) "that there exist
infinitely many sequences of three primes in arithmetic progression."
[Ribenboim's _The Little Book of Big Primes_].  If (p,q,r) is such a prime
triplet, then A047949(q) <> -1.

Charles, it appears that there's a typo in "a(n) is n-2 infinitely often,
...".  Did you possibly mean "n-3"?  If so, is that known to be true?  This
also raises the general question (pardon me if I'm overlooking something
obvious; e.g., I haven't seen that actual proof):
For any given odd prime p, do there exist infinitely many sequences of three
(not necessarily consecutive) primes in arithmetic progression?  In
particular, is, say, A063908 an infinite sequence?  (Incidentally, note that
A071781 is a subsequence of A063908.)

Finally, are there interesting sequences yet to be submitted of numbers (or
primes) not containing any arithmetic progression of three primes?  (I
haven't looked.)

Rick (will be away from e-mail for awhile now)
On Sat, Mar 13, 2010 at 5:38 PM, Jonathan Post <jvospost3 at gmail.com> wrote:

> It looks to me as if no "-1" value is there in the first 10,000
> values.  Is that correct?  Is there a proof that no such value exists?
> A047949         a(2) = a(3) = 0; for n >= 4, a(n) = largest m such that n-m
> and n+m are both primes, or -1 if no such m exists.
> What is a(n) asymptotically?
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