Prime Factor

A prime factor is a factor that is prime, i.e., one that cannot itself be factored. In general, a prime factorization takes the form

(1)

where are prime factors and are their orders. Prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.

The following table gives the prime factorization for the positive integers .

1	1	11	11	21		31	31	41	41
2	2	12		22		32		42
3	3	13	13	23	23	33		43	43
4		14		24		34		44
5	5	15		25		35		45
6		16		26		36		46
7	7	17	17	27		37	37	47	47
8		18		28		38		48
9		19	19	29	29	39		49
10		20		30		40		50

The number of not necessarily distinct prime factors of a number is denoted (Hardy and Wright 1979, p. 354) or . Conway et al. (2008) coined the term "multiprimality of " to describe , with semiprimes then being termed biprimes, numbers with three factors terms triprimes, etc. The number of prime factors is given in terms of the prime factorization above by

(2)

The first few values for , 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, ... (OEIS A001222). is plotted above up to (left) and (right). The function is implemented in the Wolfram Language as PrimeOmega[n],

The function defined by is known as the Liouville function.

The number of distinct prime factors of a number is denoted (Hardy and Wright 1979, p. 354), or sometimes or , and is implemented in the Wolfram Language as PrimeNu[n].

For example, has a single distinct prime factor, so , but two total prime factors, so .

An asymptotic series for is given by

(3)

(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where is a constant related to the Mertens constant and are Stieltjes constants. Furthermore, the variance is given by

(4)

where

			(5)
			(6)

(OEIS A091589), and

(7)

(OEIS A086242; Finch 2003) is a convergent prime sum. The coefficients and are given by the sums

			(8)
			(9)
			(10)
			(11)
			(12)

(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where

			(13)
			(14)
			(15)
			(16)

(Finch 2003).

Similarly, if is chosen at random between 1 and , then the probability that

(17)

approaches

(18)

as (Knuth 1998, p. 384). In addition, the average value of for approaches (Erdős and Kac 1940; Hardy and Wright 1979; Knuth 1998, p. 384)

The average order of is

(19)

(Hardy 1999, p. 51). More precisely,

(20)

for appropriate constants and (Hardy and Ramanujan 1917; Hardy and Wright 1979, p. 355; Hardy 1999, p. 57), where is asymptotic notation.