Almost Prime
A number n with prime factorization
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
![画像: 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],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/AlmostPrime/NumberedEquation2.svg){kind=link}
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.
See also
Chen's Theorem, Prime Factor, Prime Number, Semiprime, Sphenic NumberExplore with Wolfram|Alpha
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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 PrimeCite this as:
Weisstein, Eric W. "Almost Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlmostPrime.html