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The
divisorial (
divisor product) of
is the
product of divisors of
, i.e. the product of all
positive integers up to
which divide
.
By analogy with the term “
phitorial” for the
coprimorial of
(a product involving
, the number of positive integers up to and coprime to
), the divisorial of
might be called the “tautorial” since the product involves
, the
number of divisors of
.
Formulae
The
divisorial of
is

τ ! (n) = Π τ (n) := i = i [i ∣n] = i [n mod i = 0], 
where
is the number of divisors of
and
[·] is the
Iverson bracket.
The following identity holds

Π τ (n) = n = 2√ n τ (n) , 
where
is the number of divisors of
. Note that
is always a square since
is odd iff
is a square.
From the
canonical prime factorization of

the canonical prime factorization of
is

with

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 such that

for some
and
[·] is the Iverson bracket.
Quasi divisorial primes
The
quasi divisorial primes are
primes such that

for some
and
[·] is the Iverson bracket.
Sequences
A007955 Divisorial of
: product of numbers up to
which are
divisors/
factors of
.

{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 such that
for some
(ordered by increasing
).

{2, 7, ...}
A?????? Numbers
such that
is prime.

{3, 4, ...}
A118370 Divisorial primes (
quasi divisorial primes):
Primes such that
for some
(ordered by increasing
).

{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
such that
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, ...}
See also
Notes
External links