A052130 a(n) is the number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity), for m sufficiently large.

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

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

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.

log(y) ~ a + b*x + c*x^2, where a=1.1422, b=0.7419, and c=-0.00035, with an r^2 of 1.0. (End)

[But what is y? - Editors, Jun 15 2021]

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 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, A126279, A126281.

Sequence in context: A295145 A151998 A308631 * A366130 A368675 A065506

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(29) from Robert G. Wilson v, Feb 21 2006

Status

approved