Bad terms in sequence A108719?

[Dean Hickerson]

Mostly to Jose Brox:

> Sequence A108719 says:
>
>  5,7,13,17,19,23,29,31,37,41,43,47,53,59,61,67...
>
> Primes which can be partitioned into a sum of distinct primes in more than
> one way.
> Presumably this consists of all primes except 2, 3 and 11 - see A000586.
> Prime p is in the sequence iff A000586(p)>1
> 5 is a member because 5 = 2+3; 19 because 19 = 3+5+11.
>
> I think that two different sequences have been merged here. One is the
> simpler "primes that can be partitioned into a sum of distinct primes",
> and the other is the same adding "in more than one way".
>
> 5 is not partitioned into a sum of distinct primes in two ways or more,
> it's just 2+3, unless you consider 0 to be a prime (this seems to be the
> rule in A000586, although n|0 for n>0), but considering p+0 to be a prime
> partition is a bit weird in a primes sequence, isn't it?
>
> Btw, I don't know either why, in A000586, a(0) = 1... I can't see 0 = p+q
> with 0<=p<>q>=0, despite you say 0 is prime or not.

I've only computed a few terms by hand, but both sequences appear correct
to me.

Jose, what you're missing is the fact that a partition of a positive number
may have only one part, and the partition of zero has zero parts.  So, for
example, 5 has 2 partitions into distinct primes, namely 5 and 2+3.  (Note
that the first partition isn't 5+0, it's just 5.)

It may seem strange to call such a thing a partition, but everything works
out better if you do.  First, if you try to formally define what a
partition is, you'd have to do some extra work to prohibit the ones with
one or zero parts.  Second, things like generating functions and recursion
formulas usually turn out to be simpler if you count partitions with one or
zero parts.

For example, here's one way to formally define a partition:  A partition of
a number n is a funtion f from the positive integers into the nonnegative
integers, such that f(k)=0 for all but finitely many values of k and such
that the sum over all k of  k*f(k)  equals n.  For each k, f(k) tells how
many times n occurs in the partition.  For example, the partition
5+4+4+4+2+2 is the function defined by f(2)=2, f(4)=3, f(5)=1, and f(k)=0
for all k except 2, 4, and 5.  With this formulation, there's nothing very
strange about the partition of 0 (f(k)=0 for all k), or the partition of
n into one part (f(n)=1, f(k)=0 for all k not equal to n).

Dean Hickerson
dean at math.ucdavis.edu