>>13856947It's not too hard to construct a perfect sequence given a smaller perfect sequence, e.g. if you have 2,3,5, then the smallest term you can append for another perfect sequence is obviously 10. I think it's then a matter of finding the number of two-term perfect sequences, then from that you can find the number of three-term perfect sequences and so on. For notation, call an n-term sequence an n-sequence.
For a 2-sequence, the smallest term could be anything greater than 1, and the largest term can be anything greater than or equal to the smallest term. Accounting for the order of terms, this gives 2C(n,2)+n possible 2-sequences.
For 3-sequences, it becomes a little trickier, but you basically want to append to the sequence an integer that is larger than all the other terms in the sequence. Call this integer c, and the other two integers in the sequence a and b, where a<=b. Then c will be such that ac>=a+b+c, so c>=(a+b)/(a-1).You should be able to just count the number of such c, be sure to account for order, and then repeat the algorithm for 4-sequences and so forth.
I can't be bothered fleshing out more details right now, but hopefully that gives you a start. If you're still stuck and no one finds something better, I might come back to it later on.
