Periods of Prime Reciprocals

I've written about the periodic decimal representations of prime number reciprocals before, specifically in a post titled Cyclic Numbers on November 23rd 2019. All prime reciprocals are periodic with periods that are of the form:$$\frac{p-1}{k}\text{ where }k=1,2,3,\dots$$What I'd never considered however, was the relative frequencies of these periods for different values of $k$. It's not difficult to generate a table (permalink) to show this information. I've considered only primes in the range up to 40,000. See Figure 1.

Figure 1

As can be seen, more than a third of primes have reciprocal periods where $k=1$. It's only when we reach $k=31$ that we find no such primes in the range up to 40000. However, if we extend the range up to one million then there are 24 primes $p$ with a period of $(p-1)/31$. These primes are (permalink):

49663, 113647, 129023, 136897, 160829, 163061, 381487, 437287, 462149, 501829, 503131, 591233, 675553, 697687, 699299, 701593, 770909, 791927, 800731, 860623, 863909, 918779, 933349, 969743

It would seem that for any arbitrary $k$ there will be an infinite series of prime numbers $p$ have a reciprocal period of $(p-1)/k$. Sometimes the initial prime number is not all that large. Take $k=101$ for example. In the range up to one million there are two primes with this property, namely 5051 and 637513. For $k=197$ there are three primes: 74861, 818339 and 931811.

For any composite number, the period of its reciprocal is the product of the periods of its prime factors divided by their greatest common divisor. For example:$$\begin{array}{rl}2077& =31\times 67\\ \text{period}(\frac{1}{31})& =15\\ \text{period}(\frac{1}{67})& =33\\ \text{gcd}(15,33)& =3\\ \text{period}(\frac{1}{2077})& =\frac{15\times 33}{3}\\ & =\frac{495}{3}\\ & =165\end{array}$$What's also of interest is a list of the first primes that are divisible by the various divisors once 1 is subtracted from them. The results are shown in Figure 2 over the range up to 100,000. Notice that there is no result for 41 in this range. Notice that for 2 and 5 a result of 1 is returned. This is because when $n$ is composite, $n$.period() returns the value 1 (permalink).

Figure 2

The first prime less 1 that is divisible by 41 is 118901 whose reciprocal has a period of 2900. Thus 41 holds the record for hosting the largest prime minus 1 divisible by numbers between 1 and 50.