Question: Partition of n into prime parts

We can say that prime p is a partition prime of n if there is at least one prime partition of n having p as least part. Example 8=3+5 so 3 is a partition prime, but 5 is not.
 

Furthermore, say that p is a singular partition prime if there is one and only one partition of n with p as least part. I am trying to find numbers n for which the set Q(n) of singular partition primes is {phi}. That is to say, if we take any prime partition of n, then there are at least two partitions associated with its least member. I find so far only two examples: 63 and 161. Clearly no such n can be prime because then n is a singular partition prime of itself (Incidentally, primes having only themselves as singular partition primes are: 2,3,7,13,23,31,41,79,101,107,149..).

I am asking for a code to compute more terms for the case Q(n)={phi}. 

Thanks in advance

David.