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A101695
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a(n) = n-th n-almost prime.
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12
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,changed
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AUTHOR
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Jonathan Vos Post, Dec 12 2004
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EXTENSIONS
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a(21)-a(30) from Robert G. Wilson v, Feb 11 2006
a(12) corrected by N. J. A. Sloane, Nov 23 2007
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STATUS
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approved
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