Sum of Prime Factors
Let 
be the sum of prime factors (with
repetition) of a number 
. 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., October 5, 2003).
Now consider iterating 
until a
fixed point (which will either be 0 or a prime) is reached. For example, 20 would
give the sequence 20, 9, 6, 5, 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).
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