

A052130


a(n) = number of numbers between 1 and 2^m with mn 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
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OFFSET

0,2


COMMENTS

a(n) = number of products of halfoddprimes <= 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^(mn), 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 m1 prime factors, 2^(m1) and 3*2^(m2), 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

BerndRainer 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



