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A078842
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Sums of the antidiagonals of the table of k-almost-primes (A078840).
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14
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.
Primes in this sequence begin: a(1) = 2, a(2) = 7, a(3) = 19, a(9) = 1583, a(22) = 12946903, a(26) = 207149213. - Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 20 2006
a(40) = 3393931105067 is also prime. - Robert G. Wilson v, Nov 24 2007.
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LINKS
| Robert G. Wilson v, Table of n, a(n) for n = 0..140.
Eric Weisstein's World of Mathematics, k-Almost Prime.
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FORMULA
| a(n)=sum(i=0, n-1, A078840(i+1, n-i))
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EXAMPLE
| a(3) = 19 = 5(3rd prime) + 6(2nd 2-almost-prime) + 8(first 3-almost-prime).
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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 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]; Table[ Sum[ AlmostPrime[k, n - k + 1], {k, n}], {n, 150}] (* Robert G. Wilson v *)
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CROSSREFS
| Cf. A078840, A078841, A078843, A078844, A078845, A078846.
Sequence in context: A140610 A152461 A100119 * A110299 A112304 A006589
Adjacent sequences: A078839 A078840 A078841 * A078843 A078844 A078845
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Dec 11 2002
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EXTENSIONS
| a(12)-a(30) from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 11 2006
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