Omega Multiplicity Diagram

Mathematics makes several distinctions among natural numbers ℕ because many functions treat certain kinds of numbers in a predictable way. For instance, the divisor counting function τ(p) = 2 iff p is prime, but τ(n) > 2 for composite numbers, and τ(n) = 1 for the empty product n = 1. The Euler totient function φ(p) = p − 1 iff p is prime, etc. Squarefree numbers are those that have only 1 instance of a distinct prime factor; 6 is squarefree since 6 = 2 × 3, but 12 is not squarefree since 12 = 2² × 3. Prime powers are numbers n = p^ε ; ε ≥ 0, a designation that includes 1 and the primes.

Let us divide the natural numbers n ∈ ℕ into 5 categories based on the number of primes p | n. We recognize that ω(n) is the number of distinct p | n, while Ω(n) is the number of p | n with multiplicity. The number n is said to be squarefree iff ω(n) = Ω(n). The number n is said to be prime iff ω(n) = Ω(n) = 1, and a prime power iff ω(n) = 1. The number n = 1, the empty product, in a category all to itself, therefore, we may hold that there are actually 4 nontrivial categories. We further distinguish numbers instead with M(n) = the largest multiplicity in n, meaning the largest exponent such that any prime power p^ε | n.

Therefore {n > 1} may be partitioned into numbers with ω(n) = 1 and those with ω(n) > 1, and then into those with M(n) = 1 and those with M(n) > 1. We might assign colors as handles for these categories, noting that the “red” (■) category ω(n) = M(n) = 1 of course corresponds with the primes.

Omega-Multiplicity Diagram

	M(n) = 1	M(n) > 1
ω(n) > 1	■ multus 8, 27, 125 A246547	■ tantus 12, 75, 216 A126706
ω(n) = 1	■ prime 2, 17, 101 A40	■ varius 6, 35, 210 A120944

Therefore we have three categories of composite numbers that we shall assign names to, so as to be able to succinctly refer to them. The color codes are what is standard in this work whenever we need to refer to the aggregate kinds of natural numbers based on distinct prime factors and maximum multiplicity.

Multus numbers (■) are composite prime powers with ω(n) > 1 but M(n) = 1 and appear in OEIS A246547. The name derives from Latin multus, “many”, since we have many copies of the same primes p | n. The smallest multus number is 4. Examples of these are 16, 49, and 625. Subsets of A246547 are the squares of primes A1248, the cubes of primes A030078, etc.

Varius numbers (■) are composite squarefree numbers with ω(n) = 1 but M(n) > 1 and appear in OEIS A120944. The name derives from Latin varius, “variegated”, since we have a diverse set of distinct primes p | n, and only one copy (multiplicity) of any p. The smallest varius number is 6. Examples include 10, 77, and 2310. Subsets of A120944 include the primorials excepting {1, 2}, that is, A120944 = A2110 \ {1, 2}, and the squarefree semiprimes A6881, which are products of 2 dissimilar primes, e.g., 6, 10, 14, 15, etc. An obvious subset is the set of even semiprimes without the number 4, i.e., A100484 \ {4}.

Finally, tantus numbers (■) are composite numbers that are neither squarefree nor prime powers with both ω(n) and M(n) exceeding 1, and are listed in OEIS A126706. The name derives from Latin tantus, “so (many)”, since we have a diverse set of distinct primes p | n, and at least one prime p appears more than once, that is, the multiplicity of p in n is greater than 1. The smallest tantus number is 12; other examples are 36, 50, and 686. The highly composite numbers excepting {1, 2, 4, 6} are tantus, as well as the superior highly composite numbers. Therefore, A4490 \ {2, 6} ⊂ A2182 \ {1, 2, 4, 6} ⊂ A126706.

It is clear that the prime powers (■■) are a superset of prime and multus numbers. Squarefree numbers (■■) are a superset of prime and varius numbers. The so-called powerful (or squareful) numbers (■■) are a superset of multus and tantus numbers and {1}. OEIS A024619 (■■) is a superset of varius and tantus numbers. The composite numbers (■■■) comprise the multus, varius, and tantus numbers.

The partitions named in this work aim to render a finer distinction to basic partitions of the composite numbers.