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divisorcount6

Divisorcount6 refers to the set of positive integers that have exactly six positive divisors. In number theory, the divisor counting function d(n) (also called tau(n)) returns how many divisors n has; divisorcount6 consists of those n for which d(n) = 6.

A standard characterization uses the prime factorization of n. If n = ∏ p_i^{a_i} with primes p_i and

Examples of six-divisor numbers include 12 = 2^2×3, 18 = 2×3^2, 20 = 2^2×5, 28 = 2^2×7, 32 = 2^5, and

Applications and related concepts: divisorcount6 is a specific case within the broader divisor function theory, useful

See also: divisor function, tau(n), highly composite numbers, prime factorization.

exponents
a_i
≥
1,
then
d(n)
=
∏
(a_i
+
1).
For
d(n)
to
equal
6,
the
possible
factorizations
of
6
as
a
product
of
integers
greater
than
1
are
6
itself
and
3×2.
This
yields
two
forms
for
n:
either
n
=
p^5
for
some
prime
p,
or
n
=
p^2
q
where
p
and
q
are
distinct
primes.
Therefore,
a
number
has
exactly
six
divisors
precisely
when
it
is
either
a
fifth
power
of
a
prime
or
the
product
of
a
square
of
a
prime
with
a
different
prime.
45
=
3^2×5.
Other
instances
are
50,
52,
63,
and
many
more.
The
set
is
infinite,
since
there
are
infinitely
many
primes
p
for
which
p^5
is
available,
and
there
are
infinitely
many
pairs
of
distinct
primes
for
which
p^2
q
is
less
than
any
given
bound.
for
studying
the
distribution
of
numbers
by
their
number
of
divisors.
It
contrasts
with
numbers
having
more
or
fewer
divisors
and
connects
to
factorization
patterns
and
multiplicative
functions.