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..140.

Eric Weisstein's World of Mathematics, k-Almost Prime.

a(n) = Sum_{i=0..n-1} A078840(i+1, n-i).

a(3) = 19 = 5 (3rd prime) + 6 (2nd 2-almost prime) + 8 (first 3-almost prime).

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}]; Plus @@@ Table[t[[n - k + 1, k]], {n, 30}, {k, n, 1, -1}] (* 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 W. Weisstein 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]; Table[ Sum[ AlmostPrime[k, n - k + 1], {k, n}], {n, 150}] (* Robert G. Wilson v, Feb 11 2006 *)

Cf. A078840, A078841, A078843, A078844, A078845, A078846.

Sequence in context: A322385 A350170 A220697 * A110299 A209400 A112304

Adjacent sequences: A078839 A078840 A078841 * A078843 A078844 A078845

Benoit Cloitre and Paul D. Hanna, Dec 11 2002

a(12)-a(30) from Robert G. Wilson v, Feb 11 2006

approved