[seqfan] Re: No "-1" value in A047949?
Charles Greathouse
charles.greathouse at case.edu
Sun Mar 14 21:50:33 CET 2010
> 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?
Thanks for catching that embarrassing mistake! I must admit that was
a mind-o, not a typo. But yes, it's true (the n-3 version); by
Dirichlet's theorem, this happens Theta(n/log^2 n) of the time in
1..n. It's a much easier question than proving A063908 infinite.
Charles Greathouse
Analyst/Programmer
Case Western Reserve University
On Sun, Mar 14, 2010 at 2:29 PM, Rick Shepherd <rlshepherd2 at gmail.com> wrote:
> 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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