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A078842 Sums of the antidiagonals of the table of k-almost primes (A078840). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..140.

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

FORMULA

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

EXAMPLE

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

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 *)

CROSSREFS

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

Sequence in context: A100119 A322385 A220697 * A110299 A209400 A112304

Adjacent sequences:  A078839 A078840 A078841 * A078843 A078844 A078845

KEYWORD

nonn

AUTHOR

Benoit Cloitre and Paul D. Hanna, Dec 11 2002

EXTENSIONS

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

STATUS

approved

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Last modified January 27 14:29 EST 2021. Contains 340467 sequences. (Running on oeis4.)