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# Divisorial

The divisorial of ${\displaystyle \scriptstyle n}$ is the product of divisors of ${\displaystyle \scriptstyle n}$, i.e. the product of all positive integers up to ${\displaystyle \scriptstyle n}$ which divide ${\displaystyle \scriptstyle n}$.

By analogy with the term phi-torial for the coprimorial of ${\displaystyle \scriptstyle n}$ (a product involving ${\displaystyle \scriptstyle \varphi (n)}$ numbers,) the divisorial of ${\displaystyle \scriptstyle n}$ might be called the tau-torial since the product involves ${\displaystyle \scriptstyle \tau (n)}$ numbers, where ${\displaystyle \scriptstyle \tau (n)}$ is the number of divisors of ${\displaystyle \scriptstyle n}$.

## Formulae

Thus the divisorial of ${\displaystyle \scriptstyle n}$ is

${\displaystyle \tau _{_{_{!}}}(n)=\Pi _{\tau }(n)\equiv \prod _{\stackrel {i=1}{i|n}}^{n}i=\prod _{i=1}^{n}i^{[i|n]}=\prod _{i=1}^{n}i^{[n{\bmod {i}}=\,0]}}$

where ${\displaystyle \scriptstyle \tau (n)}$ is the number of divisors of ${\displaystyle \scriptstyle n}$ and ${\displaystyle \scriptstyle [\cdot ]}$ is the Iverson bracket.

The following identity holds

${\displaystyle \Pi _{\tau }(n)=n^{\frac {\tau (n)}{2}}={\sqrt {n^{\tau (n)}}}}$

where ${\displaystyle \scriptstyle \tau (n)}$ is the number of divisors of ${\displaystyle \scriptstyle n}$.

Note that ${\displaystyle \scriptstyle n^{\tau (n)}}$ is always a square since ${\displaystyle \scriptstyle \tau (n)}$ is odd iff ${\displaystyle \scriptstyle n}$ is a square.

From the canonical prime factorization of ${\displaystyle \scriptstyle n}$

${\displaystyle n=\prod _{i=1}^{\omega (n)}{p_{i}}^{\alpha _{i}}}$

the canonical prime factorization of ${\displaystyle \scriptstyle \tau _{_{_{!}}}(n)}$ is

${\displaystyle \tau _{_{_{!}}}(n)=\prod _{i=1}^{\omega (n)}{p_{i}}^{\beta _{i}},}$

with

${\displaystyle \beta _{i}={\frac {\alpha _{i}}{2}}\tau (n)={\frac {\alpha _{i}}{2}}\prod _{j=1}^{\omega (n)}(\alpha _{j}+1).}$

## Properties

All terms of this sequence occur only once.[1]

## Divisorial primes

The set of divisorial primes is the union of almost divisorial primes and quasi divisorial primes.

### Almost divisorial primes

The almost divisorial primes are primes ${\displaystyle \scriptstyle p}$ such that

${\displaystyle p={\Bigg (}\prod _{d|n}d{\Bigg )}-1={\Bigg (}\prod _{i=1}^{n}d^{[d|n]}{\Bigg )}-1}$

for some ${\displaystyle \scriptstyle n}$ and ${\displaystyle \scriptstyle [\cdot ]}$ is the Iverson bracket.

### Quasi divisorial primes

The quasi divisorial primes are primes ${\displaystyle \scriptstyle p}$ such that

${\displaystyle p={\Bigg (}\prod _{d|n}d{\Bigg )}+1={\Bigg (}\prod _{i=1}^{n}d^{[d|n]}{\Bigg )}+1}$

for some ${\displaystyle \scriptstyle n}$ and ${\displaystyle \scriptstyle [\cdot ]}$ is the Iverson bracket.

## Sequences

Divisorial of ${\displaystyle \scriptstyle n}$: product of numbers up to ${\displaystyle \scriptstyle n}$ which are divisors/factors of ${\displaystyle \scriptstyle n,\ n\geq \,1,}$ (A007955) gives

{1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 10077696, 37, 1444, 1521, 2560000, 41, ...}

Almost divisorial primes: Primes ${\displaystyle \scriptstyle p}$ such that ${\displaystyle \scriptstyle p=\,{\big (}\prod _{d|n}d{\big )}-1}$ for some ${\displaystyle \scriptstyle n}$ (ordered by increasing ${\displaystyle \scriptstyle n}$.)

{2, 7, ...}

Numbers ${\displaystyle \scriptstyle n}$ such that ${\displaystyle \scriptstyle {\big (}\prod _{d|n}d{\big )}-1}$ is prime.

{3, 4, ...}

Divisorial primes (quasi divisorial primes): Primes ${\displaystyle \scriptstyle p}$ such that ${\displaystyle \scriptstyle p=\,{\big (}\prod _{d|n}d{\big )}+1}$ for some ${\displaystyle \scriptstyle n}$ (ordered by increasing ${\displaystyle \scriptstyle n}$.) (A118370)

{2, 3, 37, 101, 197, 331777, 677, 8503057, 9834497, 5477, 59969537, 8837, 17957, 21317, 562448657, 916636177, 42437, 3208542737, 3782742017, 5006411537, 7676563457, 98597, 106277, 11574317057, ...}

Numbers ${\displaystyle \scriptstyle n}$ such that ${\displaystyle \scriptstyle {\big (}\prod _{d|n}d{\big )}+1}$ is prime. (A118369)

{1, 2, 6, 10, 14, 24, 26, 54, 56, 74, 88, 94, 134, 146, 154, 174, 206, 238, 248, 266, 296, 314, 326, 328, 374, 378, 386, 430, 442, 466, 472, 488, 494, 498, 510, 568, 582, ...}