0,2

A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.

Robert G. Wilson v, Table of n, a(n) for n = 0..228.

Eric Weisstein's World of Mathematics, Almost Prime.

Conjecture: Lim as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007.

a(0) = 1 since one is the multiplicative identity,

a(1) = 2nd 1-almost prime is the second prime number = A000030(2) = 3,

a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.

a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.

a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},

a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v *) ( or *)

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric Weisstein (eww(AT)wolfram.com) Feb 07 2006 *)

AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n-1, n]], {n, 30}]; lst (* Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 13 2007 *)

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

Sequence in context: A026566 A147356 A147416 this_sequence A147387 A146267 A151420

Adjacent sequences: A078838 A078839 A078840 this_sequence A078842 A078843 A078844

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2002

a(14)-a(29) from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 11 2006