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# Divisorial

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The divisorial (divisor product) of
 n
is the product of divisors of
 n
, i.e. the product of all positive integers up to
 n
which divide
 n
. By analogy with the term “phi-torial” for the coprimorial of
 n
(a product involving
 φ (n)
, the number of positive integers up to and coprime to
 n
), the divisorial of
 n
might be called the “tau-torial” since the product involves
 τ (n)
, the number of divisors of
 n
.

## Formulae

The divisorial of
 n
is
τ ! (n)  =  Πτ (n)  :=
 n ∏ i   = 1i ∣n

i  =
 n ∏ i   = 1

i [i ∣n]  =
 n ∏ i   = 1

i [n mod i = 0],
where
 τ (n)
is the number of divisors of
 n
and [·] is the Iverson bracket.

The following identity holds

Πτ (n)  =  n
 τ (n) 2
=
2  nτ (n)
,
where
 τ (n)
is the number of divisors of
 n
. Note that
 n τ (n)
is always a square since
 τ (n)
is odd iff
 n
is a square.
From the canonical prime factorization of
 n
n  =
 ω (n) ∏ i   = 1

piαi,
the canonical prime factorization of
 τ ! (n)
is
τ ! (n)  =
 ω (n) ∏ i   = 1

piβi,

with

βi  =
 αi 2
τ (n)  =
 αi 2
 ω (n) ∏ j   = 1

(αj + 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
 p
such that
p  =
 d ∣n ∏ d ∣n

d
− 1  =
 n ∏ i   = 1

d  [d ∣n]
− 1
for some
 n
and [·] is the Iverson bracket.

### Quasi divisorial primes

The quasi divisorial primes are primes
 p
such that
p  =
 d ∣n ∏ d ∣n

d
+ 1  =
 n ∏ i   = 1

d  [d ∣n]
+ 1
for some
 n
and [·] is the Iverson bracket.

## Sequences

A007955 Divisorial of
 n
: product of numbers up to
 n
which are divisors/factors of
 n, n   ≥   1
.
{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, ...}
A?????? Almost divisorial primes: Primes
 p
such that
p = (
d ∣n

 d ∣n
d )  −  1
for some
 n
(ordered by increasing
 n
).
{2, 7, ...}
A?????? Numbers
 n
such that
(
d ∣n

 d ∣n
d )  −  1
is prime.
{3, 4, ...}
A118370 Divisorial primes (quasi divisorial primes): Primes
 p
such that
p = (
d ∣n

 d ∣n
d )  +  1
for some
 n
(ordered by increasing
 n
).
{2, 3, 37, 101, 197, 331777, 677, 8503057, 9834497, 5477, 59969537, 8837, 17957, 21317, 562448657, 916636177, 42437, 3208542737, 3782742017, 5006411537, 7676563457, 98597, 106277, 11574317057, ...}
A118369 Numbers
 n
such that
(
d ∣n

 d ∣n
d )  +  1
is prime.
{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, ...}