[seqfan] Re: Seq. needed: linear comb. of primes

[Neil Sloane]

Franklin, Alois
Thanks for those helpful comments! I just created A215036,
which includes Franklin's proof - hope that is OK.
Neil

On Sun, Aug 5, 2012 at 4:28 PM, <franktaw at netscape.net> wrote:

> Just a sketch:
>
> If S_n is the set of possible values for the first n primes, then S_{n+1}
> = S_n U (S_n + prime(n+1)) U (S_n - prime(n+1)). Beyond about n=4, this
> will be everything even or everything odd in an interval around zero, and
> then a fringe on either side; the size of the interval will be 2 *
> A007504(n) - k for some small k. Recursively, since prime(n) << A007504(n),
> this will continue to hold. Hence the sequence continues to alternate 0's
> and 1's.
>
> A quite modest estimate on the distribution of primes suffices to complete
> the proof.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Neil Sloane <njasloane at gmail.com>
>
> Dear Seqfans,
> take the first n primes and combine them with coefficients +1 and -1;
> a(n) is the smallest number (in absolute value) that can be obtained.
> I get by hand (for n>=1)
> 2 1 0 1 0 1 0 1 0 1 0 1
> How does it continue? Does 1,0 repeat for ever?
> E.g. a(3) = 0 from -2-3+5
>
> a(11)=0 from 2-3-5-7+11-13+17+19-23-29+31= 0
>
> A214912 (which I just rescued from the "waiting to be submitted" stack)
> is an upper bound, but has a(11)=2 so is a different sequence.
>
> This must be a classical problem! Richard?
>
> Neil
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
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-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com