[seqfan] Splitting algorithm(s)

[hv at crypt.org]

Consider a formalized version of the splitting algorithm, used to generate
a set of distinct unit fractions that sum to a given rational:

Given a multiset M, if it has no duplicated elements, terminate; else, pick
a duplicated element e, and replace one copy of e with the two elements
e+1 and e(e+1).

If we start with the multiset { 1^n }, how many elements are in the resulting
set? I get a sequence starting: 1, 4, 16, 172, 4331 (not in OEIS), but my
naive algorithm quickly runs out of memory. Starting instead with { 2^n }
I get: 1, 3, 7, 15, 46, 171, 785, 4330, 28607 (also not in OEIS).

This multiset -> multiset operation gives a multiset -> set algorithm.
Are there other operations of the form "replace a duplicate e with the
values (f_1(e), f_2(e), ... f_k(e))" that a) are guaranteed to terminate,
and b) have useful interpretations or interesting results?

Hugo