The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$. 
Its roots have a curious onion-like structure. Cutting at $n=100$
(corresponding to $p_{100}=541$) yields a polynomial of degree $541-100=441$
with complex roots forming the picture
[![enter image description here][1]][1]

The onion-like structure of these roots is even more apparent
when taking logarithms of all non-zero roots. Logarithms of roots
in the open upper half-plane form the picture
[![enter image description here][2]][2]

In order to check that primes lead to especially nice roots, one can also
consider the series
$$\sum_{n=3}^\infty \left(\lfloor n/\log n)\rfloor-\lfloor (n-1)/\log n-1\rfloor\right)\frac{z^{n-\lfloor n/\log n\rfloor}}{n!}$$
with roots for the polynomial of degree roughly 440 forming the picture
[![enter image description here][3]][3]

which lacks the onion-structure.

Logarithms of roots in the upper half-plane give rise to
[![enter image description here][4]][4]


The series $\sum_{n=1}^\infty \frac{z^{p_n}}{n!}$ seems (perhaps) also a bit
less interesting. Cutting at $n=100$, we get the roots

[![enter image description here][5]][5]

with logarithms of roots in the upper half-plane given by

[![enter image description here][6]][6]

The onion-structure of the initial series
could of course be an artefact due to rounding errors or
due to truncation of the series at $n=100$.
I can exclude rounding errors with reasonable confidence
since I did the computations with a precision of 1500 digits.
The onion-structure of root-sets seems to persist by
setting the cutoff $n$ to higher values.

A somewhat arbitrary convention is the indexing of primes.
Series of the form $\sum_{n=1}^\infty\frac{z^{p_n-(n+\tau)}}{(n+\tau)!}$
for small $\tau=1$ or $\tau=-1$ behave however similarly. (The onion-structure
is slightly messier for $\tau=-1$ and roughly equivalent for $\tau=1$.)

Is there an explanation for this phenomenon?

Obvious remark: Shells of the limit onion-structure (in case of existence)
are finite (since roots of holomorphic
functions are discrete in their domain of holomorphicity).


  [1]: https://i.sstatic.net/l9C15.gif
  [2]: https://i.sstatic.net/DBLGP.gif
  [3]: https://i.sstatic.net/x0OX3.gif
  [4]: https://i.sstatic.net/IPowB.gif
  [5]: https://i.sstatic.net/CStWG.gif
  [6]: https://i.sstatic.net/EqyrJ.gif