Let $d(n)$ represent the total number of divisors of $n$ including negative integers. Find all the natural numbers such that
$$\large d(d(\ldots d(n)))=n$$

where the composition is arbitrarily finite

Kainui's derivation for lower bound of $n$ :
https://www.overleaf.com/read/rvgfgxkwbmnc

Question

Let $d(n)$ represent the total number of divisors of $n$ including negative integers. Find all the natural numbers such that 
$$\large d(d(\ldots d(n)))=n$$

where the composition is arbitrarily finite

Kainui's derivation for lower bound of $n$ : 
https://www.overleaf.com/read/rvgfgxkwbmnc

ganeshie8 · Answer

I found one solution by trial and error : 
d(8) = 8
d(d(8)) = 8
...

dan815 · Answer

there coule be a multiple repetition case

dan815 · Answer

4 is another one

dan815 · Answer

there could be cases like 

alternating like maybe something like 6 to 8 to 6 to 8

ganeshie8 · Answer

4 doesn't work

dan815 · Answer

oh my bad 4 is also a divisor xD

dan815 · Answer

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