TOPICS
Prime Omega Function
The prime omega function is the arithmetic function on the positive integers given by
 |
(1)
|
when the prime factorization of 
is
 |
(2)
|
where the 
are distinct primes and the 
are positive integers.
Thus 
counts prime factors with multiplicity,
and 
.
It is implemented in the Wolfram Language
as PrimeOmega[n].
The first few values of 
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). For example,
because 
.
In contrast, the prime nu function gives 
because it ignores multiplicity.
Conway et al. (2008) called 
the "multiprimality" of 
, and accordingly used the names biprime for a semiprime,
triprime when 
,
and so on.
The prime omega function is completely additive,
 |
(3)
|
for all positive integers 
and 
. It determines the Liouville
function by
 |
(4)
|
For a positive integer 
, let
 |
(5)
|
An asymptotic series for this mean is
 |
(6)
|
where ![B_2=B_1+sum_(k=1)^(infty)1/[p_k(p_k-1)]](https://mathworld.wolfram.com/images/equations/PrimeOmegaFunction/Inline17.svg)
,
is the Mertens constant, 
is the 
th prime, and the 
are Stieltjes constants
(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003). The corresponding variance
over the integers 1, 2, ..., 
has expansion
 |
(7)
|
where
 |  |  |
(8)
|
 |  |  |
(9)
|
(OEIS A091589; Finch 2003, pp. 6-7), and
 |
(10)
|
(OEIS A086242; Finch 2003) is a convergent prime sum. The coefficients are
 |  |  |
(11)
| ||
 |  |  |
(12)
| ||
 |  |  |
(13)
| ||
 |  | ![-gamma_1-(gamma-1)[gamma-2sum_(k=1)^(infty)(lnp_k)/((p_k-1)^2)]](https://mathworld.wolfram.com/images/equations/PrimeOmegaFunction/Inline40.svg) |  |  |
(14)
|
 |  |  |
(15)
|
Equivalently, the sums occurring here have values
 |  |  |
(16)
|
 |  |  |
(17)
|
(Finch 2003).
If 
is chosen uniformly from 1 through 
, then the probability that
 |
(18)
|
approaches
 |
(19)
|
as 
(Erdős and Kac 1940; Knuth 1998, p. 384). This is the Erdős-Kac
theorem for 
.
The average order of 
is 
(Hardy 1999, p. 51). More precisely,
 |
(20)
|
where 
is asymptotic notation (Hardy and Ramanujan
1917; Hardy and Wright 1979, p. 355; Hardy 1999, p. 57).
See also
Asymptotic Notation, Erdős-Kac Theorem, Liouville Function, Mertens Constant, Prime Factor, Prime Nu Function, Prime Sums, Stieltjes ConstantsExplore with Wolfram|Alpha
References
Conway, J. H.; Dietrich, H.; and 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. 62, 738-742, 1940.Finch,
S. "Two Asymptotic Series." Dec. 10, 2003. https://web.archive.org/web/20160419150604/http://www.people.fas.harvard.edu/~sfinch/csolve/asym.pdf.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. "The Normal Number
of Prime Factors of a Number 
." Quart. J. Math. 48, 76-92, 1917.Hardy,
G. H. and Wright, E. M. §§22.10-22.11 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 354-355, 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,
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.
Cite this as:
Weisstein, Eric W. "Prime Omega Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimeOmegaFunction.html