The sum of prime factors (with repetition) of a positive integer 
,
denoted 
,
is obtained by adding its prime factors. The empty
sum gives 
. For example, 
, so 
. Then 
for 
, 2, ... is given by 0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7,
13, 9, 8, ... (OEIS A001414). The sum of prime
factors function is also known as the integer logarithm.
The high-water marks are 0, 2, 3, 4, 5, 7, 11, 13, 17, ..., which occur at positions 1, 2, 3, 4, 5, 7, 11, 13, 17, ... (OEIS A046022), which, with the exception of the first term, correspond exactly to the actual values of the high-water marks.
If 
is considered to be 0 for 
a prime, then the sequence
of high-water marks is 0, 4, 5, 6, 7, 9, 10, 13,
15, 19, 21, 25, 31, 33, ... (OEIS A088685),
which occur at positions 1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, ... (OEIS
A088686). Rather amazingly, if the first 7
terms are dropped, then the last digit of the high-water
marks 
and the last digit of their positions 
fall into one of the four patterns 
, (3, 2), (5, 6), or (9, 4) (A. Jones, pers.
comm., Oct. 5, 2003).
For iteration, return to the original 
function and extend it by
.
Starting from a positive integer, form the finite
iteration list ending at the
first fixed point, included once. The fixed
point will be 0, 4, or a prime. For example,
20 gives the list 20, 9, 6, 5. The fixed
points for 
,
2, ... are then given by 0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, ... (OEIS A029908),
and the lengths of the corresponding sequences are 2,
1, 1, 1, 1, 2, 1, 3, 3, 2, 1, 2, 1, 4, ... (OEIS A002217).
Now consider the restricted sums of the iteration lists after discarding the initial term. For example, 20 would
give 
.
Then the only numbers less than 
that are equal to the sums of their
restricted iteration lists
are 20, 38, and 74.
The similar function 
giving the sum of distinct prime factors of 
can also be considered. For 
, 2, ..., this function has
the values 0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, ... (OEIS A008472).
