TOPICS
Search

Almost Prime


A number n with prime factorization

 n=product_(i=1)^rp_i^(a_i)

is called k-almost prime if it has a sum of exponents sum_(i=1)^(r)a_i=k, i.e., when the prime factor (multiprimality) function Omega(n)=k.

The set of k-almost primes is denoted P_k.

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 k-almost primes less than or equal to n 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],

and so on, where pi(x) is the prime counting function and p_k is the kth 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 k-almost primes for small k.

nOEISn-almost primes
1A0000402, 3, 5, 7, 11, 13, ...
2A0013584, 6, 9, 10, 14, 15, 21, 22, ...
3A0146128, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, ...
4A01461316, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ...
5A01461432, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, ...

See also

Chen's Theorem, Prime Factor, Prime Number, Semiprime, Sphenic Number

Explore with Wolfram|Alpha

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."

Referenced on Wolfram|Alpha

Almost Prime

Cite this as:

Weisstein, Eric W. "Almost Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlmostPrime.html

Subject classifications