If $\varphi$ denotes the Euler totient and $n=p_1^{k_1}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ we have

$ \varphi(n)= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})= p_1^{k_1} \left(1- \frac{1}{p_1} \right) p_2^{k_2} \left(1- \frac{1}{p_2} \right) \cdots p_r^{k_r} \left(1- \frac{1}{p_r} \right)$.

I using a function defined by $\psi(n) = \varphi(p_1^{k_1}) + \varphi(p_2^{k_2})+ \cdots + \varphi(p_r^{k_r})$.

I am wondering if this function has a "classical name" that I should use (and maybe another well-known notation than $\psi$).

Thank you!

  • 1
    $\begingroup$ You could try calculating it for $n=1,2,\dots,10$, say, and then looking up the resulting sequence in the Online Encyclopedia of Integer Sequences. If it's there, it might have a name, reference to places people have used it, etc. $\endgroup$ – Gerry Myerson Mar 11 '12 at 3:39
  • $\begingroup$ Good advice Gerry, thanks, the first 13 are 1,1,2,2,4,3,6,4,6,5,10,4,13 and OEIS doesn't find any match. $\endgroup$ – Nathan Portland Mar 11 '12 at 12:35

I doubt this function has a classical name or notation. It is an example of an "additive function", which means that $\psi(ab) = \psi(a) + \psi(b)$ whenever $\gcd(a,b)=1$.

  • $\begingroup$ Thanks Greg for this point of view $\endgroup$ – Nathan Portland Mar 10 '12 at 22:58

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