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

Author

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

Keywords

• nonn
• nice

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, Feb 01 2000

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]

Examples

			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.
		

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

Crossrefs

Cf. A117630, A126279, A126281.

Programs

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 *)

Python

from math import prod, isqrt
from sympy import primepi, primerange, integer_nthroot
def A052130(n):
    if n<=1: return n+1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    k = 1
    while 3**k<(r:=1<Chai Wah Wu, Dec 03 2024

Extensions

More terms from David W. Wilson, Feb 01 2000

a(24)-a(29) from Robert G. Wilson v, Feb 21 2006