The Arithmetic Derivative
MATL, 12 bytes

|1&gt;?GtYf/s}0


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Explanation

Consider an integer a with |a|>1, and let the (possibly repeated) prime factors of |a| be f1, ..., fn. Then the desired result is a·(1/f1 + ... + 1/fn).

|1&gt;     % take input's absolute value. Is it greater than 1?
?       % if so:
  Gt    %   push input twice
  Yf    %   prime factors. For negative input uses its absolute value
  /     %   divide element-wise
  s     %   sum of the array
}       % else:
  0     %   push 0