# Are there names for any of these four classes of numbers related to divisors and totatives?

Are there names for any of these four classes of numbers related to divisors and totatives?

1. A [insert name here] of $n$ is a positive integer $\leq n$ that isn't a divisor of $n$ and that can be multiplied by some other integer to produce a power of $n$.
2. A [insert name here] of $n$ is a positive integer $\leq n$ that can be multiplied by some other integer to produce a power of $n$. (This class of numbers is composed of the previous class of numbers plus the divisors.)
3. A [insert name here] of $n$ is a positive integer $\leq n$ that is the product of at least one prime divisor of $n$ and at least one prime totative of $n$.
4. A [insert name here] of $n$ is a positive integer $\leq n$ that is either a totative of $n$ or the product of at least one prime divisor of $n$ and at least one prime totative of $n$. (This class of numbers is composed of the previous class of numbers plus the totatives.)

For example, for $n=18$ here are all the numbers in all the four classes:

1. {4, 8, 12, 16}
2. {1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
3. {10, 14, 15}
4. {5, 7, 10, 11, 13, 14, 15, 17}

I've done some research, and I found two articles about two classes of numbers that are somewhat similar to class number 2:

• The class of regular numbers according to Wikipedia is like class number 2, except that this class is restricted to $n=30k$ (because they are used in the study of the Babylonian's sexagesimal numeral system ) and the numbers are not restricted to being $\leq n$.
• The class of regular numbers according to Wolfram MathWorld is like class number 2, except that this class is restricted to $n=10k$ (because they are used in the study of our decimal numeral system) and the numbers are not restricted to being $\leq n$.

## 2 Answers

OEIS tells me that 1. is the set of semidivisors (https://oeis.org/A243822), 2. is the set of semidivisors or divosors (naturally) and has no idea, how could be 3. and 4. named. I have assumed in my reasearch that you are only interested in products of two primes, if that is not the case please correct me.

1. In Hardy and Wright's classic number theory text, they define your (1) as the set of semidivisors. This is not restricted to the products of two primes.

2. Similarly, (2) should be called the set of divisors and semidivisors.

3. Your (3) should be termed the set of "semitotatives" both in analogy with (1) and as in this OEIS note.

4. Finally, your (4) should be called the set of totatives and semitotatives.