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.

REFERENCES:

Conway, J. H.; Dietrich, H.; O'Brien, E. A. "Counting Groups: Gnus, Moas, and Other Exotica." Math. Intell. 30, 6-18, 2008.

Diaconis, P. "Asymptotic Expansions for the Mean and Variance of the Number of Prime Factors of a Number ." Dept. Statistics Tech. Report 96, Stanford, CA: Stanford University, 1976.

Diaconis, P. "G. H. Hardy and Probability???" Bull. London Math. Soc. 34, 385-402, 2002.

Erdős, P. and Kac, M. "The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions." Amer. J. Math. 26, 738-742, 1940.

Finch, S. "Two Asymptotic Series." December 10, 2003. https://algo.inria.fr/bsolve/.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Hardy, G. H. and Ramanujan, S. Quart. J. Math. 48, 76-92, 1917.

Hardy, G. H. and Wright, E. M. §22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 384, 1998.

Knuth, D. E. Selected Papers on Analysis of Algorithms. Stanford, CA: CSLI Publications, pp. 338-339, 2000.

Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 327, 2000.

Sloane, N. J. A. Sequences A001222/M0094, A001221/M0056, A030059, A083342, A086242, and A091589 in "The On-Line Encyclopedia of Integer Sequences."

Turán, P. "On a Theorem of Hardy and Ramanujan." J. London Math. Soc. 9, 274-276, 1934.

Turán, P. "Über einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan." J. London Math. Soc. 11, 125-133, 1936.