After a longer break I try to find factors of that numbers, taking the database of Markus Tervooren as source of needed information.

A smaller part of my finds is reported to Kurt Beschorner already.

The remaining part is collected in this plain text file (Aug 15, 2023).

New finds are collected here (Aug 20, 2023).

Newer finds are collected here (Sep 10, 2023) and here (Nov 23, 2023).

Latest finds (Nov 30, 2023).

A short note.
People maintaining lists of factorizations of $10^n+1$-numbers take care of so called $L$- and $M$-values
useing equations such as $10^{20k-10}+1 = (10^{4k-2}+1)$ · $L_{20k-10}$ · $M_{20k-10}$, where
$L_{20k-10}=A_{20k-10}-10^k$ · $B_{20k-10}$ and $M_{20k-10}=A_{20k-10}+10^k$ · $B_{20k-10}$ with
$A_{20k-10}=10^{8k-4}+5$ · $10^{6k-3}+7$ · $10^{4k-2}+5$ · $10^{2k-1}+1$ and
$B_{20k-10}=10^{6k-3}+2$ · $10^{4k-2}+2$ · $10^{2k-1}+1$.
But I will not do so. For my purposes it is not necessary and not helpful.

Playing with the database, I detected many factors which were uploaded by anonymous finders.
In case of interest I collected the factors in this plain text file (Aug 15, 2023).
More finds by Anonymous are collected here (Aug 20, 2023) and here (Sep 10, 2023).

My finds (not reported to Kurt Beschorner already) are collected in this plain text file (Aug 15, 2023).

New finds are collected here (Aug 20, 2023).

Newer finds are collected here (Sep 10, 2023) and here (Nov 23, 2023).

Latest finds (Dec 7, 2023).

Again I detected many factors which were uploaded by anonymous finders.
In case of interest I collected the factors in this plain text file (Aug 15, 2023), too. More finds by Anonymous are collected here (Sep 10, 2023).

10^1599-1: p55=2862577930401391395910619889316394633711512313018581507 (Aug 18, 2023; sigma=6089901937320443; B1=11000000; B2=93177894810)

In rare cases it luckily happens that after finding a factor the remaining cofactor is prime.
If this cofactor is large, then its primality is not obviously but has to be proved.
In some cases I did this using the program primo of Marcel Martin, see the table below.
The corresponding certificates are uploaded to and verified by the database.