A058067 is multiplicative

[David Wilson]

For prime p, form sequence s_p in which n appears with the the multiplicity 
of p in the prime factorization of n, e.g:

   s_2 = 2 4 4 6 8 8 8 10 12 12 14 16 16 16 16 18 20 20 ...
   s_3 = 3 6 9 9 12 15 18 18 21 24 27 27 27 30 33 36 36 ...
   s_5 = 5 10 15 20 25 25 30 35 40 45 50 50 55 60 65 70 ...
   etc.

8 appears 3 times in s_2 because 2 has exponent 3 in the prime factorization 
of 8.

Let t_p be the running sum of s_p beginning with t_p(0) = 0:

   t_2 = 0 2 6 10 16 24 32 40 50 62 74 88 104 120 136 152  ...
   t_3 = 0 3 9 18 27 39 54 72 90 111 135 162 189 216 246 ...
   t_5 = 0 5 15 30 50 75 100 130 165 205 250 300 350 405 ...
   etc.

Then A058067 is multiplicative with a(p^e) = p^t_p(e).

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- David Wilson