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# 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, ...

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

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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."

Almost Prime

## Cite this as:

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