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
$$\sum_{n=3}^\infty \left(\lfloor n/\log n)\rfloor-\lfloor (n-1)/\log n-1\rfloor\right)\frac{z^{n-\lfloor n/\log n\rfloor}}{\lfloor n\log n\rfloor!}$$
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).

**Added:** Computations using 800 digits and ending with $p_{250}=1583$ yield pictures

[![enter image description here][7]][7]

for roots of $\sum_{n=1}^{250} z^{p_n-n}/n!$.
There is a shell of roots and a few 'isolated' roots outside the domain of convergency.

Logarithms of roots in the upper open halfplane and in the convergency domain are given by

[![enter image description here][8]][8]

For the polynomial $\sum_{n=1}^{250}z^{p_n}/n!$
roots (computed again with Digits set to 800 in Maple) are

[![enter image description here][9]][9]

There are again a few roots outside of the convergency domain but there seems to be 
an additional shell of roots (artefact due to insufficient precision or real roots?).

Logarithms of roots in the convergency domain and in the upper open halfplane are given by

[![enter image description here][10]][10]

SECOND ADDITION: The root-set of $$\sum_{n=1}^\infty \frac{z^n}{\phi(n)!}$$
(which is an entire function if I am not mistaken)
with $\phi(n)$ denoting the Euler totient function counting invertible classes
modulo $n$ is even more spectacular. Cutting at $n=840$, the set of 
roots is given by

[![enter image description here][11]][11]

In this case, there is however an easy non-rigourous explanation:
The Euler totient function is small for friable integers. 
Taking the factorial of $\phi(n)$ makes coefficients of non-friable 
integers so small that they are almost invisible to roots. (By the way,
this series should always be truncated with leading coefficient
a friable integer involving all small primes in 
order to avoid spurious 'ghost'-roots.) 
Roots of this series
correspond therefore to a lacunary series with non-zero coefficients
given by suitable friable integers. Roots of lacunary series
with rapidly decaying coefficients
have onion-structures with shells corresponding to pairs of
consecutive non-zero coefficients.
This is probably the explanation of the nice onion structure in this
case (this explanation does however not apply for the series 
involving primes considered above).


  [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
  [7]: https://i.sstatic.net/EYwnU.gif
  [8]: https://i.sstatic.net/QIxh1.gif
  [9]: https://i.sstatic.net/ijenT.gif
  [10]: https://i.sstatic.net/sejFW.gif
  [11]: https://i.sstatic.net/DFdMV.gif