## Divisor

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

 (1)

For any divisor of , where
 (2)

so
 (3)

Now, , so there are possible values. Similarly, for , there are possible values, so the total number of divisors of is given by
 (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
 (5)

so
 (6)

and
 (7)

The Geometric Mean of divisors is
 (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
 (9)

 (10)

For a given ,
 (11)

Summing over all ,
 (12)

so . Splitting and into prime factors,
 (13)

For a prime Power , the divisors are 1, , , ..., , so
 (14)

For , therefore,
 (15)

For the special case of a Prime, (15) simplifies to
 (16)

For a Power of two, (15) simplifies to
 (17)

The Arithmetic Mean is
 (18)

The Harmonic Mean is
 (19)

But , so and
 (20)

and we have
 (21)

 (22)

Given three Integers chosen at random, the probability that no common factor will divide them all is
 (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,

 (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