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 A101695 a(n) = n-th n-almost prime. 12
 2, 6, 18, 40, 108, 224, 480, 1296, 2688, 5632, 11520, 25600, 53248, 124416, 258048, 540672, 1105920, 2228224, 4587520, 9830400, 19922944, 40894464, 95551488, 192937984, 396361728, 822083584, 1660944384, 3397386240, 6845104128 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity. This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. This is the diagonal just below A078841. LINKS Robert G. Wilson v and Charles R Greathouse IV, Table of n, a(n) for n = 1..1000 (first 229 terms from Robert G. Wilson) Eric Weisstein's World of Mathematics, Almost Prime. FORMULA Conjecture: lim_{ n->inf.} a(n+1)/a(n) = 2. - Robert G. Wilson v, Oct 07 2007, Nov 13 2007 Stronger conjecture: a(n)/(n * 2^n) is polylogarithmic in n. That is, there exist real numbers b < c such that (log n)^b < a(n)/(n * 2^n) < (log n)^c for large enough n. Probably b and c can be chosen close to 0. - Charles R Greathouse IV, Aug 28 2012 EXAMPLE a(1) = first 1-almost prime = first prime = A000040(1) = 2. a(2) = 2nd 2-almost prime = 2nd semiprime = A001358(2) = 6. a(3) = 3rd 3-almost prime = A014612(3) = 18. a(4) = 4th 4-almost prime = A014613(4) = 40. a(5) = 5th 5-almost prime = A014614(5) = 108. MAPLE A101695 := proc(n)     local s, a ;     s := 0 ;     for a from 2^n do         if numtheory[bigomega](a) = n then             s := s+1 ;             if s = n then                 return a;             end if;         end if;     end do: end proc: # R. J. Mathar, Aug 09 2012 MATHEMATICA 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]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n, n]], {n, 30}]; lst (* Robert G. Wilson v, Oct 07 2007 *) CROSSREFS Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606. Sequence in context: A034881 A146345 A064842 * A014741 A016059 A027556 Adjacent sequences:  A101692 A101693 A101694 * A101696 A101697 A101698 KEYWORD nonn,changed AUTHOR Jonathan Vos Post, Dec 12 2004 EXTENSIONS a(21)-a(30) from Robert G. Wilson v, Feb 11 2006 a(12) corrected by N. J. A. Sloane, Nov 23 2007 STATUS approved

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