

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
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



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



KEYWORD

nonn,nice


AUTHOR

BerndRainer Lauber (br.lauber(AT)surf1.de), Jan 21 2000


EXTENSIONS



STATUS

approved



