Sequence of occurrences in Euler $\phi$ function?

Question

Let $a_n$ denote the number of positive integers $m$ such that $\phi(m) = n$, where $\phi$ denotes Euler's totient function. The first several terms of the sequence, excluding zeros, are given by

$$\{a_n\} = \{2,3,4,4,5,2,6,6,4,5,2,10,2,\dotsc\}.$$

I would like to continue the sequence to $1,\!000$ terms.

Q: What are the first thousand terms of the sequence in comma separated form? Does this sequence have any interesting properties?

Answer 1

Yes, it has interesting properties:

• $\varphi(a\cdot b)= \gcd(a,b)\cdot \varphi({a\over\gcd(a,b)})\cdot \varphi({b\over\gcd(a,b)})$

Being the most important here. It allows conclusions like:

• if $c$ is odd, then $2c$ has the same totient value as $c$
• if $c$ is even, and $c$ is not divisible by 3, then $2c$ has the same totient value as $3c$
• if $c$ is divisible by 6, but not divisible by 5, then $4c$ has the same totient value as $5c$
• etc.