Do they exist? I look for both pairs and one number of a pair.

Yes, and there is a lot of them. Smallest examples:

X44 Walker&Einstein 2001
5480828320492525=5^2*7^2*11*13*19*31*17*23*103*1319
5786392931193875=5^3*7*11*13*19*31*37*43*61*809

X43 Chernykh 2015
445953248528881275=3^2*5^2*7*13*19*37*43*73*439*22483
659008669204392325=5^2*7*13*19*37*73*571*1693*5839

Would it be possible to extract from the database a list of the current lowest pairs not divisible by 2 or 3
(and maybe additionally by 2, 3 or 5). For the latter (2 ,3 and 5) there was a paper years back with an example of 30 digits
or more but maybe this has been improved

For anyone interested I found the first of these (the X44 pair) many years back when I was doing a search with David Einstein
for 14 and 15 digit pairs, we have sure progressed! At the time Jan Pederson made a suggestion to me it
would be interesting to see if the program we were using could find any of these. Well it did find this pair very quickly!

I hope that in the future the boinc project can extend the depth for these types much further!

Andrew

PS This all gets a mention by Mariano Garcia in "Amicable Pairs, A Survey" third page

https://ir.cwi.nl/pub/10756/10756D.pdf

Here's the list: https://sech.me/ap/coprime_to_6.txt. It's very easy to find such pairs with the current method of search, I just went through all factorization ranges starting with 5^N for up to 24-digit numbers. One PC was enough for it.