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
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, ... |
REFERENCES:
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."
