Let be the sum of prime factors (with repetition) of a number n. For example, , so . Then for n = 1, 2, ... is given by 0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, ... (Sloane's 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, ... (Sloane's 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 p a prime, then the sequence of high-water marks is 0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, ... (Sloane's A088685), which occur at positions 1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, ... (Sloane's A088686). Rather amazingly, if the first 7 terms are dropped, then the last digit of the high-water marks a and the last digit of their positions b 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 n = 1, 2, ... are then given by 0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, ... (Sloane's A029908), and the lengths of the corresponding sequences are 2, 1, 1, 1, 1, 2, 1, 3, 3, 2, 1, 2, 1, 4, ... (Sloane's 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 10⁶ 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 n can also be considered. For n = 1, 2, ..., this function has the values 0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, ... (Sloane's A008472).

 

Prime Factor, Prime Sums