Multiplicative partition

A multiplicative partition (also called disordered factorization ) of a natural number is a way of representing this number as a product of natural numbers greater than . Two factorizations are the same if each factor of one factorization also occurs in the other and they only differ in their order. The number itself is also viewed as a partition of itself. Multiplicative partitions have been researched since 1923 at the latest , but at that time under the Latin name “ factorisatio numerorum ”. The current name probably came about from an article published in 1983 by Jeffrey Shallit and John F. Hughes in the journal " American Mathematical Monthly " on the subject. ${\displaystyle n>1}$ ${\displaystyle 1}$ ${\displaystyle n}$

Examples

The number 20 has 4 multiplicative partitions, namely . ${\displaystyle 20=2\cdot 10=4\cdot 5=2\cdot 2\cdot 5}$

The number 30 has 5 multiplicative partitions, namely . The number 30 is square-free . ${\displaystyle 30=2\cdot 15=3\cdot 10=5\cdot 6=2\cdot 3\cdot 5}$

The number 81 has 5 multiplicative partitions, namely . The number 81 can be represented as a prime power: ${\displaystyle 81=3\cdot 27=9\cdot 9=3\cdot 3\cdot 9=3\cdot 3\cdot 3\cdot 3}$ ${\displaystyle 3^{4}}$

The number 109 has only one multiplicative partition, namely itself. It is also a prime number.

number

Let be the number of all multiplicative partitions of . The first values ​​of are: ${\displaystyle a_{n}}$ ${\displaystyle n}$ ${\displaystyle a_{n}}$

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, ... sequence A001055 in OEIS

Percy Alexander MacMahon and A. Oppenheim noted that the Dirichlet series generating function also has the following product representation: ${\displaystyle f(s)}$

${\displaystyle f(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}=\prod _{k=2}^{\infty }{\frac {1}{1-k^{-s}}}}$

Special cases

Is square-free - i.e. does not contain a prime number more than once in the prime factorization , or , where stands for the Möbius function - then is the number of multiplicative partitions , where the -th is Bell's number and the number of unique prime factors of . ${\displaystyle n}$ ${\displaystyle \mu (n)\neq 0}$ ${\displaystyle \mu (n)}$ ${\displaystyle B_{\omega (n)}}$ ${\displaystyle B_{i}}$ ${\displaystyle i}$ ${\displaystyle \omega (n)}$ ${\displaystyle n}$

The second special case assumes that the number is the result of a power with a prime number as a base and a natural exponent . Formally: ${\displaystyle n}$

${\displaystyle \exists p\in \mathbb {P} ~\exists m\in \mathbb {N} :n=p^{m}}$

Where stands for the set of all prime numbers. This precondition can also be noted as congruence : ${\displaystyle \mathbb {P} }$

${\displaystyle \exists p\in \mathbb {P} :n\equiv 0\mod p}$

If one of these conditions is met - if it is one, so is the other automatically - then the number of possible multiplicative partitions is the same as the additive partition of the exponent . This is clear because the prime factorization is too. ${\displaystyle m}$

The third special case is the most trivial. It assumes that it is itself a prime number, so that is true. Due to the definition of prime numbers, there can only be one factorization, namely itself. ${\displaystyle n}$ ${\displaystyle n\in \mathbb {P} }$ ${\displaystyle n}$

application

In their article published in 1983, Jeffrey Shallit and John F. Hughes described an application of multiplicative partitions to classify natural numbers using the number of divisors . For example:

${\displaystyle \forall p\in \mathbb {P} ~\forall q\in \mathbb {P} \setminus \left\{p\right\}~\forall r\in \mathbb {P} \setminus \left\{p,q\right\}:\sigma _{0}(p^{11})=\sigma _{0}(p^{5}\cdot q)=\sigma _{0}(p^{3}\cdot q^{2})=\sigma _{0}(p^{2}\cdot q\cdot r)=12}$

Where , and - as formalized - are pairwise different prime numbers, where is the number function and where would be the divisor function . This example was constructed from the multiplicative partitions . ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle r}$ ${\displaystyle \sigma _{0}}$ ${\displaystyle \sigma _{k}}$ ${\displaystyle 12=2\cdot 6=3\cdot 4=2\cdot 2\cdot 3}$

In general, for every multiplicative partition of with factors (where a factor is for ) ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle t_{i}}$ ${\displaystyle 1\leq i\leq k}$

${\displaystyle n=\prod _{i=1}^{k}t_{i}}$

there are associated with them a lot of natural numbers with exact divisors. Each of these numbers has the shape ${\displaystyle n}$

${\displaystyle \prod _{i=1}^{k}p_{i}^{t_{i}-1}}$,

where all are pairwise different prime numbers. ${\displaystyle p_{i}}$

Individual evidence

1. ^ " The American Mathematical Monthly> Vol. 90, No. 7, Aug.-Sep., 1983> On the Number of Multiplicative Partitions ". Retrieved May 19, 2014