is called -almost
prime if it has a sum of exponents, i.e., when the prime
factor (multiprimality) function .
The set of -almost
primes is denoted .
The primes correspond to the "1-almost prime" numbers and the 2-almost prime numbers correspond to semiprimes.
Conway et al. (2008) propose calling these numbers primes, biprimes, triprimes,
and so on.
Formulas for the number of -almost
primes less than or equal to are given by
and so on, where
is the prime counting function and is the th prime (R. G. Wilson V, pers. comm., Feb. 7,
2006; the first of which was discovered independently by E. Noel and G. Panos
around Jan. 2005, pers. comm., Jun. 13, 2006).
The following table summarizes the first few -almost primes for small .
Conway, J. H.; Dietrich, H.; O'Brien, E. A. "Counting Groups: Gnus, Moas, and Other Exotica." Math. Intell.30, 6-18,
2008.Sloane, N. J. A. Sequences A000040/M0652,
A001358/M3274, A014612,
A014613, and A014614
in "The On-Line Encyclopedia of Integer Sequences."
with prime factorization
-almost
prime if it has a sum of exponents 
, i.e., when the prime
factor (multiprimality) function 
.
-almost
primes is denoted 
.
-almost
primes less than or equal to 
are given by
![pi^((2))(n)=sum_(i=1)^(pi(n^(1/2)))[pi(n/(p_i))-i+1],
pi^((3))(n)=sum_(i=1)^(pi(n^(1/3)))sum_(j=i)^(pi((n/p_i)^(1/2)))[pi(n/(p_ip_j))-j+1],
pi^((4))(n)=sum_(i=1)^(pi(n^(1/4)))
sum_(j=i)^(pi((n/p_i)^(1/3)))sum_(k=j)^(pi((n/(p_ip_j))^(1/2)))[pi(n/(p_ip_jp_k))-k+1],](/images/equations/AlmostPrime/NumberedEquation2.svg)
is the prime counting function and 
is the 
th prime (R. G. Wilson V, pers. comm., Feb. 7,
2006; the first of which was discovered independently by E. Noel and G. Panos
around Jan. 2005, pers. comm., Jun. 13, 2006).
-almost primes for small 
.
-almost primes