The distinct prime factors of a positive integer 
are defined as the 
numbers 
, ..., 
in the prime
factorization
 |
(1)
|
(Hardy and Wright 1979, p. 354).
A list of distinct prime factors of a number 
can be computed in the Wolfram
Language using FactorInteger[n][[All,
1]], and the number 
of distinct prime factors is implemented as PrimeNu[n].
The first few values of 
for 
, 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2,
1, 1, 2, 1, 2, ... (OEIS A001221; Abramowitz
and Stegun 1972, Kac 1959). This sequence is given by the inverse Möbius
transform of 
,
where 
is the characteristic function of the
prime numbers (Sloane and Plouffe 1995, p. 22). The prime factorizations and
distinct prime factors of the first few positive integers are listed in the table
below.
 | prime factorization |  | distinct prime factors (A027748) |
| 1 | -- | 0 | -- |
| 2 | 2 | 1 | 2 |
| 3 | 3 | 1 | 3 |
| 4 |  | 1 | 2 |
| 5 | 5 | 1 | 5 |
| 6 |  | 2 | 2, 3 |
| 7 | 7 | 1 | 7 |
| 8 |  | 1 | 2 |
| 9 |  | 1 | 3 |
| 10 |  | 2 | 2, 5 |
| 11 | 11 | 1 | 11 |
| 12 |  | 2 | 2, 3 |
| 13 | 13 | 1 | 13 |
| 14 |  | 2 | 2, 7 |
| 15 |  | 2 | 3, 5 |
| 16 |  | 1 | 2 |
The numbers consisting only of distinct prime factors are precisely the squarefree numbers.
A sum involving 
is given by
 |
(2)
|
for 
(Hardy and Wright 1979, p. 255).
The average order of 
is
 |
(3)
|
(Hardy 1999, p. 51). More precisely,
 |
(4)
|
(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where 
is the Mertens constant
and 
are Stieltjes constants. Furthermore, the
variance is given by
 |
(5)
|
where
 |  |  |
(6)
|
 |  |  |
(7)
|
(OEIS A091588), where
 |
(8)
|
(OEIS A085548) is the prime zeta function 
(Finch 2003). The coefficients 
and 
are given by the sums
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  | ![-gamma_1-(gamma-1)[gamma+2sum_(k=1)^(infty)(lnp_k)/(p_k(p_k-1))]+2sum_(k=1)^(infty)((2p_k-1)(lnp_k)^2)/(2p_k(p_k-1)^2)](/images/equations/DistinctPrimeFactors/Inline47.svg) |
(12)
|
 |  |  |
(13)
|
(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
(Finch 2003).
If 
is a primorial, then
 |
(18)
|
(Hardy and Wright 1979, p. 355).
The summatory function of 
is given by
 |
(19)
|
where 
is the Mertens constant (Hardy 1999, p. 57),
the 
term (Hardy and Ramanujan 1917; Hardy and Wright 1979, p. 355) has been rewritten
in a more explicit form, and 
and 
are asymptotic notation.
The first few values of the summatory function
are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (OEIS A013939).
In addition,
![sum_(k=2)^n[omega(k)]^2=n(lnlnn)^2+O(nlnlnn)](/images/equations/DistinctPrimeFactors/NumberedEquation9.svg) |
(20)
|
(Hardy and Wright 1979, p. 357).
The first few numbers 
which are products of an odd number of distinct prime factors
(Hardy 1999, p. 64; Ramanujan 2000, pp. xxiv and 21) are 2, 3, 5, 7, 11,
13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, ... (OEIS A030059).
satisfies
![sum_(n=1)^infty1/(u_n^s)=1/2[(zeta(s))/(zeta(2s))-zeta(s)]](/images/equations/DistinctPrimeFactors/NumberedEquation10.svg) |
(21)
|
(Hardy 1999, pp. 64-65). In addition, if 
is the number of 
with 
, then
 |
(22)
|
(Hardy 1999, pp. 64-65).
." 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.Finch, S. "Two
Asymptotic Series." December 10, 2003. 
." Quart. J. Math. 48, 76-92, 1917.Hardy,
G. H. and Wright, E. M. "The Number of Prime Factors of 
" and "The Normal Order of 
and 
." §22.10 and 22.11 in 
and 
." Cited by Finch (2003). Unpublished note,
2003.Ramanujan, S.