Inspired by Andrei's nice solution of 52609, (namely `consecutive`

perfect polynomials):
Denote by $A$ the full ring of polynomials in one variable $t$ over the finite field with $2$ elements.
For any polynomial $P \in A$ define
$$
\sigma(P) = \sum_{d \mid A} d.
$$
A polynomial $P \in A$ is called `perfect`

if
$$
P = \sigma(P).
$$
If $P$ has no roots in $GF(2)$ then $P$ is `odd`

, otherwise it is `even`

.

Andrei proved:

$P$ is odd perfect iff $P$ is perfect and $P$ is a square in $A,$ and deduced:

$P$ and $P+1$ cannot both be perfect.

Observe that one of $P,P+1$ is odd while the other is even.

Take now a polynomial $P \in A.$ What is the `next`

polynomial of the same `type`

?
i.e., having the same `parity`

. Seems that the following definition is appropriate for this:

Call `neighbors`

two polynomials $P,Q \in A$ if $\deg(P)>2$ and ($Q=P+t(t+1)$ or $P=Q+t(t+1)$).

Question: There are neighbors polynomials $P,Q \in A$ such that $P$ and $Q$ are both perfect ?