Almost Prime
A number 
with prime
factorization
 |
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
![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.gif) |
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 
.
 | OEIS |  -almost primes |
| 1 | A000040 | 2, 3, 5, 7, 11, 13, ... |
| 2 | A001358 | 4, 6, 9, 10, 14, 15, 21, 22, ... |
| 3 | A014612 | 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, ... |
| 4 | A014613 | 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... |
| 5 | A014614 | 32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, ... |
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