0,2

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

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. This is the diagonal just below A078841.

Robert G. Wilson v, Table of n, a(n) for n = 0..229.

Eric Weisstein's World of Mathematics, Almost Prime.

Conjecture: lim_{ n->inf.} a(n+1)/a(n) = 2. - Robert G. Wilson v, Oct 07 2007, Nov 13 2007

a(1) = first 1-almost prime = first prime = A000040(1) = 2.

a(2) = 2nd 2-almost prime = 2nd semiprime = A001358(2) = 6.

a(3) = 3rd 3-almost prime = A014612(3) = 18.

a(4) = 4th 4-almost prime = A014613(4) = 40.

a(5) = 5th 5-almost prime = A014614(5) = 108.

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, n]], {n, 30}]; lst (* Robert G. Wilson v (rgwv@rgwv.com), Oct 07 2007 *)

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606.

Sequence in context: A066286 A034881 A064842 this_sequence A014741 A016059 A027556

Adjacent sequences: A101692 A101693 A101694 this_sequence A101696 A101697 A101698

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 12 2004

a(21)-a(30) from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 11 2006

a(12) corrected by njas, Nov 23 2007