Divisor
المؤلف:
Dickson, L. E.
المصدر:
History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.
الجزء والصفحة:
...
26-6-2020
2089
Divisor
A divisor, also called a factor, of a number 
is a number 
which divides 
(written 
). For integers, only positive divisors are usually considered, though obviously the negative of any positive divisor is itself a divisor. A list of (positive) divisors of a given integer 
may be returned by the Wolfram Language function Divisors[n].
Sums and products are commonly taken over only some subset of values that are the divisors of a given number. Such a sum would then be denoted, for example,
 |
(1)
|
Such sums are implemented in the Wolfram Language as DivisorSum[n, form, cond].
The following tables lists the divisors of the first few positive integers (OEIS A027750).
 |
divisors |
| 1 |
1 |
| 2 |
1, 2 |
| 3 |
1, 3 |
| 4 |
1, 2, 4 |
| 5 |
1, 5 |
| 6 |
1, 2, 3, 6 |
| 7 |
1, 7 |
| 8 |
1, 2, 4, 8 |
| 9 |
1, 3, 9 |
| 10 |
1, 2, 5, 10 |
| 11 |
1, 11 |
| 12 |
1, 2, 3, 4, 6, 12 |
| 13 |
1, 13 |
| 14 |
1, 2, 7, 14 |
| 15 |
1, 3, 5, 15 |
The total number of divisors for a given number 
(variously written 
, 
, or 
) can be found as follows. Write a number in terms of its prime factorization
 |
(2)
|
For any divisor 
of 
, 
where
 |
(3)
|
so
 |
(4)
|
Now, 
, so there are 
possible values. Similarly, for 
, there are 
possible values, so the total number of divisors 
of 
is given by
 |
(5)
|
The product of divisors can be found by writing the number 
in terms of all possible products
 |
(6)
|
so
and
 |
(10)
|
The geometric mean of divisors is
The arithmetic mean is
 |
(14)
|
The harmonic mean is
 |
(15)
|
But 
, so 
and
and we have
 |
(19)
|
 |
(20)
|
Given three integers chosen at random, the probability that no common factor will divide them all is
![[zeta(3)]^(-1) approx 1.20206^(-1) approx 0.831907,](https://mathworld.wolfram.com/images/equations/Divisor/NumberedEquation12.gif) |
(21)
|
where 
is Apéry's constant.
The smallest numbers having exactly 0, 1, 2, ... divisors (other than 1) are 1, 2, 4, 6, 16, 12, 64, 24, 36, ... (OEIS A005179; Minin 1883-84; Grost 1968; Roberts 1992, p. 86; Dickson 2005, pp. 51-52). Fontené (1902) and Chalde (1903) showed that if 
is the prime factorization of the least number with a given number of divisors, then (1) 
is prime, (2) 
is prime except for the number 
which has 8 divisors (Dickson 2005, p. 52).
Let 
be the number of elements in the greatest subset of ![[1,n]](https://mathworld.wolfram.com/images/equations/Divisor/Inline56.gif)
such that none of its elements are divisible by two others. For 
sufficiently large,
 |
(22)
|
(Le Lionnais 1983, Lebensold 1976/1977).
REFERENCES:
Chalde. Nouv. Ann. Math. 3, 471-473, 1903.
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, 2005.
Fontené, G. Nouv. Ann. Math. 2, 288, 1902.
Grost, M. E. "The Smallest Number with a Given Number of Divisors." Amer. Math. Monthly 75, 725-729, 1968.
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.
Minin, A. P. Math. Soc. Moscow 11, 632, 1883-84.
Nagell, T. "Divisors." §1 in Introduction to Number Theory. New York: Wiley, pp. 11-12, 1951.
Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., 1992.
Sloane, N. J. A. Sequences A005179/M1026 and A027750 in "The On-Line Encyclopedia of Integer Sequences."
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