A078842 Sums of the antidiagonals of the table of k-almost primes (A078840).

1, 2, 7, 19, 44, 95, 195, 395, 794, 1583, 3172, 6334, 12665, 25313, 50596, 101180, 202326, 404635, 809227, 1618410, 3236766, 6473474, 12946903, 25893723, 51787365, 103574668, 207149213, 414298342, 828596584, 1657193052, 3314385970

Offset: 0

Author

Benoit Cloitre and Paul D. Hanna, Dec 11 2002

Keywords

• nonn

Comments

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

Examples

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..140.
• Eric Weisstein's World of Mathematics, k-Almost Prime.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

Extensions

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