A071810

[franktaw at netscape.net]

The numbers representable as the sum of distinct values from any finite 
set are symmetric: if a set S sums to m, the complement of S sums to N 
- m.  So what I'm claiming is that every number up to N is 
representable as the sum of primes up to Prime(n), except for 1, 4, 6, 
11, N-11, N-6, N-4, and N-1.

This doesn't so much follow from "every number except 1, 4, 6, and 11 
is representable as the sum of distinct primes" (like I said) as the 
other way around.  You show this by a fairly straightforward if 
slightly messy induction; I'm not going to go into the details.

Franklin T. Adams-Watters


-----Original Message-----
From: maxale at gmail.com

  Franklin,

 It is not clear to me:
 1) why each prime <=N is representable as the sum of distinct primes
 <= Prime(n).
 2) why the primality of N-11 (for example) affects the result.

 Thanks,
 Max

 On 9/7/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
 > Every number except 1, 4, 6, and 11 is representable as the sum of
  > distinct primes. So taking N = A007504(n) = sum_{k=1}^n Prime(k), 
for
  > n >= 4 (N >= 28 > 2*11), this is just PrimePi(N) - 1 - isprime(N-1) 
-
 > isprime(N-4) - isprime(N-6) - isprime(N-11).
 >
 > (Here isprime is A010051, the characteristic function of primes.)
 >
 > Franklin T. Adams-Watters
 >
 >
 > -----Original Message-----
 > From: maxale at gmail.com
 >
 > On 9/7/06, Max A. <maxale at gmail.com> wrote:
 >
  > > The reason why this does not eat a lot of memory is the fact that 
the
 > > set of all numbers representable as the sum of some of the first n
 > > primes is rather small.
 >
 > Suggestion to a new sequence: a(n) = the number of primes
  > representable as the sum of some subset of the set of first n 
primes.
 > Is it in OEIS?
 > A071810 counts exactly these primes with multiplicities (i.e.,
 > counting different representations of the same prime separately).
 >
 > Max
 >
 >
  > 
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