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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 “
phi-torial” for the
coprimorial of
(a product involving
, the number of positive integers up to and coprime to
), the divisorial of
might be called the “tau-torial” 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