Let sopfr(n) be the sum of prime factors (with repetition) of a number n. For example, 20 = 2^2·5, so sopfr(20) = 2 + 2 + 5 = 9. Then sopfr(n) for n = 1, 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.
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