Partitions Derived From Prime-Subset

[Leroy Quet]

I will just copy/paste my reply from the sci.math thread at:

http://groups.google.com/groups?dq=&hl=en&lr=&ie=UTF-8&threadm=40502848.204
0009%40cox.net&prev=


I wonder here on seq.fan about the sequence formed from the number of 
distinct partitions.

Leroy

--


I wrote on sci.math (original post and reply):
> Let P be any subset of the primes.
> (example: P contains all primes of even-index,
> or those congruent to 1 (mod 6).)
> 
> Let A be the set of positive integers: including 1 and every positive
> integer which is a multiple of only the primes in P, and A  contains
> no member which is divisible by a prime not in P.
> 
> Let B be the set of positive integers: including 1 and every positive
> integer which is a multiple of only the primes *not* in P, and B
> contains no member which is divisible by a prime *in* P.
> 
> (Yes, there are positive integers in neither set, as long as P does
> not contain every prime.)
> 
> 
> Let g(x) be the number of distinct elements of A which are <= x,
> for x = a positive real.
> 
> 
> So then, for m = any positive integer:
> 
> ---
> \
> /    g(m/k)
> ---
> k=elements of B, k <= m
> 
> always = m.
> 
> 
> In linear-mode:
> 
> sum{k=elements of B, k<=m}  g(m/k)
> 
> always equals m.
> 
> 
> (Right?)
> 
> thanks,
> Leroy Quet


Every sum of g(m/k)'s (written of most recently above)
represents a partition of m (where each g is a part in the partition).

I wonder how many *distinct* partitions of m are represented by sums of 
g's.
(Each g-sum corresponds with a given subset P of the primes <= m.)

I get by-hand that the sequence begins:

1, 2, 3, 4, 5, 7,..

thanks,
Leroy Quet