A divisor of a number
is a number
which Divides
, also called a Factor. The total number of
divisors for a given number
can be found as follows. Write a number in terms of its Prime Factorization
![\begin{displaymath}
N={p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdots{p_r}^{\alpha_r}.
\end{displaymath}](d2_1214.gif) |
(1) |
For any divisor
of
,
where
![\begin{displaymath}
d={p_1}^{\delta_1}{p_2}^{\delta_2}\cdots {p_r}^{\delta_r},
\end{displaymath}](d2_1216.gif) |
(2) |
so
![\begin{displaymath}
d'={p_1}^{\alpha_1-\delta_1}{p_2}^{\alpha_2-\delta_2}\cdots {p_r}^{\alpha_r-\delta_r}.
\end{displaymath}](d2_1217.gif) |
(3) |
Now,
, so there are
possible values. Similarly, for
, there are
possible values, so the total number of divisors
of
is given by
![\begin{displaymath}
\nu(N)=\prod_{n=1}^r (\alpha_n+1).
\end{displaymath}](d2_1223.gif) |
(4) |
The function
is also sometimes denoted
or
.
The product of divisors can be found by writing
the number
in terms of all possible products
![\begin{displaymath}
N=\cases{d^{(1)}d'^{(1)}\cr \vdots\cr d^{(\nu)}d'^{(\nu)}\cr},
\end{displaymath}](d2_1226.gif) |
(5) |
so
and
![\begin{displaymath}
\prod d = N^{\nu(N)/2}.
\end{displaymath}](d2_1230.gif) |
(7) |
The Geometric Mean of divisors is
![\begin{displaymath}
G\equiv \left({\prod d}\right)^{1/\nu(N)} = [N^{\nu(n)/2}]^{1/\nu(N)} = \sqrt{N}.
\end{displaymath}](d2_1231.gif) |
(8) |
The sum of the divisors can be found as follows. Let
with
and
. For any divisor
of
,
, where
is a divisor of
and
is a divisor of
. The divisors of
are 1,
,
,
..., and
. The divisors of
are 1,
,
, ...,
. The sums of the divisors are then
![\begin{displaymath}
\sigma(a)=1+a_1+a_2+\ldots+a
\end{displaymath}](d2_1240.gif) |
(9) |
![\begin{displaymath}
\sigma(b)=1+b_1+b_2+\ldots+b.
\end{displaymath}](d2_1241.gif) |
(10) |
For a given
,
![\begin{displaymath}
a_i(1+b_1+b_2+\ldots+b)=a_i\sigma(b).
\end{displaymath}](d2_1242.gif) |
(11) |
Summing over all
,
![\begin{displaymath}
(1+a_1+a_2+\ldots+a)\sigma(b)=\sigma(a)\sigma(b),
\end{displaymath}](d2_1243.gif) |
(12) |
so
. Splitting
and
into prime factors,
![\begin{displaymath}
\sigma(N)=\sigma({p_1}^{\alpha_1})\sigma({p_2}^{\alpha_2})\cdots\sigma({p_r}^{\alpha_r}).
\end{displaymath}](d2_1245.gif) |
(13) |
For a prime Power
, the divisors are 1,
,
, ...,
, so
![\begin{displaymath}
\sigma({p_i}^{\alpha_i})=1+p_i+{p_i}^2+\ldots+{p_i}^{\alpha_i} ={{p_i}^{\alpha_i+1}-1\over p_i-1}.
\end{displaymath}](d2_1248.gif) |
(14) |
For
, therefore,
![\begin{displaymath}
\sigma(N) = \prod_{i=1}^r {{p_i}^{\alpha_i+1}-1\over p_i-1}.
\end{displaymath}](d2_1249.gif) |
(15) |
For the special case of
a Prime, (15) simplifies to
![\begin{displaymath}
\sigma(p)={p^2-1\over p-1}=p+1.
\end{displaymath}](d2_1250.gif) |
(16) |
For
a Power of two, (15) simplifies to
![\begin{displaymath}
\sigma(2^\alpha) = {2^{\alpha+1}-1\over 2-1} = 2^{\alpha +1}-1.
\end{displaymath}](d2_1251.gif) |
(17) |
The Arithmetic Mean is
![\begin{displaymath}
A(N)\equiv {\sigma(N)\over \nu(N)}.
\end{displaymath}](d2_1252.gif) |
(18) |
The Harmonic Mean is
![\begin{displaymath}
{1\over H}\equiv {1\over N}\left({\sum {1\over d}}\right).
\end{displaymath}](d2_1253.gif) |
(19) |
But
, so
and
![\begin{displaymath}
\sum {1\over d}={1\over N}\sum d' = {1\over N} \sum d = {\sigma(N)\over N},
\end{displaymath}](d2_1255.gif) |
(20) |
and we have
![\begin{displaymath}
{1\over H(N)} = {1\over \nu(N)} {\sigma(N)\over N} = {A(N)\over N}
\end{displaymath}](d2_1256.gif) |
(21) |
![\begin{displaymath}
N=A(N)H(N).
\end{displaymath}](d2_1257.gif) |
(22) |
Given three Integers chosen at random, the probability that no common factor will divide them all is
![\begin{displaymath}[\zeta(3)]^{-1}\approx 1.20206^{-1} \approx 0.831907,
\end{displaymath}](d2_1258.gif) |
(23) |
where
is Apéry's Constant.
Let
be the number of elements in the greatest subset of
such that none of its elements are divisible by
two others. For
sufficiently large,
![\begin{displaymath}
0.6725\ldots \leq {f(n)\over n} \leq 0.673\ldots
\end{displaymath}](d2_1262.gif) |
(24) |
(Le Lionnais 1983, Lebensold 1976/1977).
See also Aliquant Divisor, Aliquot Divisor, Aliquot Sequence, Dirichlet Divisor Problem,
Divisor Function, e-Divisor, Exponential Divisor, Greatest Common Divisor, Infinary
Divisor, k-ary Divisor, Perfect Number, Proper Divisor, Unitary Divisor
References
Guy, R. K. ``Solutions of
.'' §B18 in
Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 73-75, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 43, 1983.
Lebensold, K. ``A Divisibility Problem.'' Studies Appl. Math. 56, 291-294, 1976/1977.
© 1996-9 Eric W. Weisstein
1999-05-24