Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer, wythagoras's Python answer, and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity.

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, either n is prime and the while loop added nothing, so add 1
             # or n.abs < 2, so return 0 anyway
             # 0**s is used in the code because it returns 1 if s == 0 and 0 for all other s
    end
  end
  return s
end

Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer, wythagoras's Python answer, and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity.

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, either n is prime and the while loop added nothing, so add 1
             # or n.abs < 2, so return 0 anyway
             # 0**s is used in the code because it returns 1 if s == 0 and 0 for all other s
    end
  end
  return s
end

Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity.

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};s+m<2?0:0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, either n is prime and the while loop added nothing, so add 1
             # or n.abs < 2, so return 0 anyway
             # 0**s is used in the code because it returns 1 if s == 0 and 0 for all other s
    end
  end
  return s
end

Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer, wythagoras's Python answer, and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity.

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, either n is prime and the while loop added nothing, so add 1
             # or n.abs < 2, so return 0 anyway
             # 0**s is used in the code because it returns 1 if s == 0 and 0 for all other s
    end
  end
  return s
end

Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity. My addition is to note that, if m = p1e1·p2e2·...·pnen, then a(m) = m·(e1/p1 + e2/p2 + ... + en/pn).

Edit: Forgot my own dang negative test cases. Sigh

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};s+m<2?0:0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};s+m<2?0:0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, because it is prime and the while loop added nothing, add 1
    end
  end
  return s
end

Ruby, 87 66 80 75 70 68 bytes

This answer is based on Luis Mendo's MATL answer and the idea that the arithmetic derivative of a number m is equal to m·(1/p1 + 1/p2 + ... + 1/pn) where p1...pn is every prime factor of n to multiplicity.

->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};m<2?0:s+0**s}

This function is called in the following way:

> a=->n{s=0;(2...m=n.abs).map{|d|(m/=d;s+=n/d)while m%d<1};s+m<2?0:0**s}
> a[299792458]
196831491

Ungolfing:

def a(n)
  s = 0
  m = n.abs
  (2...m).each do |z|
    while m%d == 0
      m /= d
      s += n / d
    end
  end
  if s == 0
    if n > 1
      s += 1 # if s is 0, either n is prime and the while loop added nothing, so add 1
             # or n.abs < 2, so return 0 anyway
             # 0**s is used in the code because it returns 1 if s == 0 and 0 for all other s
    end
  end
  return s
end