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# Divisorial

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

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

## Formulae

Thus the divisorial of $\scriptstyle n$ is

$\tau_{_{_{!}}}(n) = \Pi_{\tau}(n) \equiv \prod_{\stackrel{i = 1}{i|n}}^{n} i = \prod_{i = 1}^{n} i^{[i|n]} = \prod_{i = 1}^{n} i^{[n \bmod i=\, 0]}$

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

The following identity holds

$\Pi_{\tau}(n) = n^{\frac{\tau(n)}{2}} = \sqrt{n^{\tau(n)}}$

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

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

From the canonical prime factorization of $\scriptstyle n$

$n = \prod_{i = 1}^{\omega(n)} {p_i}^{\alpha_i}$

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

$\tau_{_{_{!}}}(n) = \prod_{i = 1}^{\omega(n)} {p_i}^{\beta_i},$

with

$\beta_i = \frac{\alpha_i}{2} \tau(n) = \frac{\alpha_i}{2} \prod_{j = 1}^{\omega(n)} (\alpha_j+1).$

## 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 $\scriptstyle p$ such that

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

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

### Quasi divisorial primes

The quasi divisorial primes are primes $\scriptstyle p$ such that

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

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

## Sequences

Divisorial of $\scriptstyle n$: product of numbers up to $\scriptstyle n$ which are divisors/factors of $\scriptstyle n,\ n\ge\, 1,$ (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 $\scriptstyle p$ such that $\scriptstyle p=\, \big( \prod_{d|n} d \big) - 1$ for some $\scriptstyle n$ (ordered by increasing $\scriptstyle n$.)

{2, 7, ...}

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

{3, 4, ...}

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