Some years ago I came up with a weird idea which I think was of my own, although it is not sensible to make such assertions.

The idea was to define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ which imitates the behaviour of the derivative of a product, i.e. (uv)' = u'v + uv'. Using the prime notation, I define the derivative of a natural number $n$ whose primary decomposition is $\; n = \prod_{i = 1}^{i = r} p_i^{\alpha_i}$ as $$n' := \sum _{j = 1}^{j = r} \alpha_j p_j^{\alpha_j - 1}\prod_{i \neq j}^{i = r} p_i^{\alpha_i} = n (\sum_{i=1}^{i = r} \frac{\alpha_i}{p_i}).$$

So I somehow applied the rule of the derivative of a multiple product, hence $(nm)' = n'm + mn'$ follows. Some particular cases are $1' = 0$ and $p' = 1$ if $p$ is prime.

Although I have never found useful this "natural derivative", I got interested on it. Here are some nice facts about the natural derivative:

  • Solutions of $n' = n$ are $n = p ^p,$ where $p$ is prime. It has an air of the exponential $e^x$.
  • The "average value" of $n'$ is $n \times \sum_{p \in \mathbb{P} }\frac{1}{p(p-1)}$
  • One can reformulate the Goldbach conjecture as: All even numbers greater than $2$ are the image of a pseudoprime number (i.e. $pq$, where $p$ and $q$ are prime).
  • It can be easily extended to $\mathbb{Q}$.

So now I have introduced to you this marvellous function, I would like to ask you:

Did I invented it or has it been used before?

In the second case: In which context does it appear, and how much is known about it?

Thanks a bunch for your answers.


1 Answer 1


Good going! There's a whole Wikipedia page on this (link), and some people have seriously considered the properties of this function. Hopefully the references there will satisfy your curiosity for a while.

Also, this sequence is listed in the Online Encyclopedia of Integer Sequences as sequence $\text{A}003415$. Usually some interesting stuff can be found there.

Finally, if you're interested in computing these numbers, see this question over on Programming Puzzles & Code Golf SE, which is how I first found out about this function.


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