

A052130


a(n) is the number of numbers between 1 and 2^m with mn prime factors (counted with multiplicity), for m sufficiently large.


7



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
From Robert G. Wilson v, Apr 13 2020 (Start):
This sequence shows a sufficiently large row of A126279 read backwards or a sufficiently large column of A126279 read vertically.
ln(y) =~ a + b*x + c*x^2, where a=1.1422, b=0.7419 & c=0.00035, with an r^2 of 1.0. (End)


LINKS

Martin Raab, Table of n, a(n) for n = 0..41 (Terms 0..35 from Robert G. Wilson v)
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, A126279, A126281.
Sequence in context: A295145 A151998 A308631 * A065506 A330454 A121165
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(29) from Robert G. Wilson v, Feb 21 2006


STATUS

approved



