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Sometime before I thought to define ....proth ( what is the correct name for sixteen? )

I think it should be called
HexadecaProth, and the next one if we ever encounter it is
IcosaProth
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It means 16 primes, but today to find such a pattern is impossible, even by a fast sieve and fast prp checking.

I don't think it's impossible! We have to find enough (i.e., many!) DodecaProths and one of them will surely be HexadecaProth. Check the following chain of 16 primes (the socalled bitwin with 7 links):
517144126484002331*13#*2^n+/1
n=0,1,2,3,4,5,6,7; and "#" denotes primorial.
composed of 8 twins, and two Cunningham chains of first and second kind. Have you ever thought that such a chain exists?
Only two such chains are currently known. The twin for n=0 has 23 digits. This is I think the place to look for HexadecaProths, around our exponent n=65 or 70.
If you can change your program to search for DodecaProths (ignoring of course potenital OctoProths whose "right legs" have small factors) I'll be willing to try!