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

%I A078842 #35 Feb 16 2025 08:32:48
%S A078842 1,2,7,19,44,95,195,395,794,1583,3172,6334,12665,25313,50596,101180,
%T A078842 202326,404635,809227,1618410,3236766,6473474,12946903,25893723,
%U A078842 51787365,103574668,207149213,414298342,828596584,1657193052,3314385970
%N A078842 Sums of the antidiagonals of the table of k-almost primes (A078840).
%C A078842 A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.
%H A078842 Robert G. Wilson v, <a href="/A078842/b078842.txt">Table of n, a(n) for n = 0..140</a>.
%H A078842 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">k-Almost Prime</a>.
%F A078842 a(n) = Sum_{i=0..n-1} A078840(i+1, n-i).
%e A078842 a(3) = 19 = 5 (3rd prime) + 6 (2nd 2-almost prime) + 8 (first 3-almost prime).
%t A078842 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 *)
%t A078842 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 *)
%t A078842 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 *)
%Y A078842 Cf. A078840, A078841, A078843, A078844, A078845, A078846.
%K A078842 nonn
%O A078842 0,2
%A A078842 _Benoit Cloitre_ and _Paul D. Hanna_, Dec 11 2002
%E A078842 a(12)-a(30) from _Robert G. Wilson v_, Feb 11 2006