Which were the latest 13 consecutive years with the most friday 13. in total? Shortest program wins, of course.
Wolfram Language 171 bytes
Any modern 13-year interval will have between 20 and 24 Friday the 13ths.
SortBy[{{#[[1,1]],#[[-1,1]]},Total[#[[2]]&amp;/@#]}&amp;/@Partition[Table[{y,Length@Cases[DateObject@{y,m,13}~Table~{m,12},x_/;DayName@x==Friday]},{y,1900,2019}],13,1],Last][[-1]]

* {{2007, 2019}, 24} *)

The above generates all the Friday the 13ths for each year between 1900 and 2019, partitions the results into intervals of 13 years, and finds the total for each such interval.  The interval of 2007-2019 has 24, the maximum number of Friday the 13ths.
If 2019 cannot be used, the correct answer will be the interval of 200-2018, which also contains 24 Friday the 13ths.