Yes, this is sequence A001913 in the online encyclopedia of integer sequences. The sequence starts 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113

Primes p such that the decimal expansion of 1/p has period p-1, which is the greatest period possible for any integer

The first few are:

1/7 = 0.142857 (6 digits)

1/ 17 = 0.0588235294117647 (16 digits)

1 / 19 = 0.052631578947368421 (18 digits)

1 / 23 = 0.0434782608695652173913 (22 digits)

1/ 29 = 0.0344827586206896551724137931 (28 digits)

1/47 = 0.0212765957446808510638297872340425531914893617 (46 digits)

1/ 59 = 0.0169491525423728813559322033898305084745762711864406779661 (58 digits)

1 / 61 = 0.016393442622950819672131147540983606557377049180327868852459 (60 digits)

1/97 = 0.010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567 (96 digits)

1/109 = 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211 (108 digits)

So try for instance 37 / 109 = 0.33944954128440366972477064220183486238532110091743119266055045871559633027522935779816513761467889908256880733944954128440366972477064220183486238532110091743119266055045871559633027522935779816513761

They are related to the Cyclic numbers and see also Repeating decimall - Cyclic numbers (in wikipedia)

Doesn’t seem that there is any algorithm for generating them (if there was I’m sure they’d say in the OEIS)

Here is an online arbitrary precision calculator you can use to test them.

Arbitrary Precision Rational Calculator