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A052130 a(n) = number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity), for m sufficiently large. 6
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; text; internal format)
OFFSET

0,2

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

m is sufficiently large precisely when 2^m > 3^(m-n), i.e., when m >= floor(n*log(3)/log(1.5)) = A117630(n+1) = A126281(n) for n > 1. (Robert G. Wilson v asks if this conjecture holds in a comment to A126281.) - David A. Corneth, Apr 09 2015

LINKS

Table of n, a(n) for n=0..29.

Index entries for sequences related to numbers of primes in various ranges

EXAMPLE

Between 1 and 2^m there is just one number with m prime factors, namely 2^m, so a(0) = 1.

For m >= 3, up to 2^m there are 2 numbers with m-1 prime factors, 2^(m-1) and 3*2^(m-2), so a(1) = 2.

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 *)

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, Feb 21 2006 *)

CROSSREFS

Cf. A117630, A126281.

Sequence in context: A209633 A216633 A151998 * A065506 A121165 A093652

Adjacent sequences:  A052127 A052128 A052129 * A052131 A052132 A052133

KEYWORD

nonn,nice

AUTHOR

Bernd-Rainer Lauber (br.lauber(AT)surf1.de), Jan 21 2000

EXTENSIONS

More terms from David W. Wilson, Feb 01 2000

a(24)-a(30) from Robert G. Wilson v, Feb 21 2006

STATUS

approved

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Last modified May 27 10:03 EDT 2017. Contains 287204 sequences.