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A052130
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For m very large, a(n) = number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity).
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6
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1, 2, 7, 15, 37, 84, 187, 421, 914, 2001, 4283, 9184, 19611, 41604, 87993, 185387, 389954, 817053, 1709640, 3567978, 7433670, 15460810, 32103728, 66567488, 137840687, 285076323, 588891185, 1215204568, 2505088087, 5159284087
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) = number of products of half-odd-primes <= 2^n. E.g. a(2) = 7 since 1, 3/2, (3/2)^2, (3/2)^3, (3/2)*(5/2), 5/2, 7/2 are all <= 2^2 - David W. Wilson (davidwwilson(AT)comcast.net).
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LINKS
| Index entries for sequences related to numbers of primes in various ranges
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EXAMPLE
| Between 1 and 2^m there is just one number with m prime factor, namely 2^m, so a(0) = 1. There are 2 numbers with m-1 prime factors (2^(m-1) and 3*2^(m-2)), so a(1) = 2.
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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 Weisstein (eww(AT)wolfram.com) Feb 07 2006
Table[ AlmostPrimePi[Floor[n(1 + 1/Sqrt@2)] + 2, 2^(n + Floor[n(1 + 1/Sqrt@2)]) + 2]], {n, 2, 30}] (* Robert G. Wilson v *)
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CROSSREFS
| Sequence in context: A095091 A131412 A151998 * A065506 A121165 A093652
Adjacent sequences: A052127 A052128 A052129 * A052131 A052132 A052133
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KEYWORD
| nonn,nice
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AUTHOR
| Bernd-Rainer Lauber (br.lauber(AT)surf1.de), Jan 21 2000
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net), Feb 01 2000.
a(24)-a(30) from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 21 2006
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