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

[franktaw at netscape.net]

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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