@Kitonum Yes I mean that {phi} is empty set and I thought that notation was universal, just that I have no way to write it as it appears in text books.

Questions:

1. No restriction on the number of summands or the starting (least) prime which I call a "partition prime" of n (my own terminology, it might be called by another name elsewhere).

2. Yes this is about partition of any whole number n into primes, so all the summands are primes and they don't have to be distinct.

3. Let P(n) be the set of partition primes of n, so for n=12, P(12)={2,3,5}. Any prime partition of 12 must have one of these numbers as least part. Q(n) is a subset of P(n) and lists the "singular partition primes of n", meaning those which appear (as least part) in one and only one partition of n into primes. So Q(12)={3,5} because 3+3+3+3 = 5+7 = 12 are the only possible choices having least part 3 and 5 respectively.

I have a recursive method for computing the overall number of partitions of n into primes by adding up the number of partitions with least part p, for every p in P(n). This works but is laborious and I don't know how to tackle a code for it, which is why I sought assistance on here.

4. The strange thing about 63 is that it has no singular partition prime. Compare the following: 46 is chosen at random:

P(46)= {2,3,5,7,11,17,23}, and the associated partition numbers are: (2,456), (3, 77), (5, 17), (7, 4), (11,1), (17,1), (23,1), where (p,m) means there are m partitions (of 46) having least part p. Thus Q(46)={11,17,23}. However P(63)={2,3,5,7,11,13,17}, for which the corresponding partition numbers (for all p in P(63)) are: (2,2198), (3,323), (5,60), (7,15), (11,5), (13,2), (17,2). Since there is no (p,1) in this collection, Q(63)={phi}, the empty set. I thought that this was a one-off anomaly number because no ´n<63 has this property, but then found that 161 has the same property. P(161)={2,3...47} and the partition numbers are (2,1305679), (3,124055), (5,15646), (7,3006), (11,687), (13,236), (17,85), (19,38), (23,16) (29,9), (31,4), (37,2), (41,4), (43,2), and (47,2). There is no (p,1) in this set so Q(161)={phi}. My calculations have now reached 167 and these are the only two numbers having this property, which is why I am curious to know if there are any more.

I hope this clarifies my question and that you can help with a code.

Regards,

David

ps: Carl please say what was wrong with my use of {phi}?