A126281 -id:A126281 - cp's OEIS frontend

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

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

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]

			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.

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

Cf. A117630, A126279, A126281.

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 *)

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

More terms from David W. Wilson, Feb 01 2000

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

A182769 Beatty sequence for (4 + sqrt(2))/2.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 18, 21, 24, 27, 29, 32, 35, 37, 40, 43, 46, 48, 51, 54, 56, 59, 62, 64, 67, 70, 73, 75, 78, 81, 83, 86, 89, 92, 94, 97, 100, 102, 105, 108, 110, 113, 116, 119, 121, 124, 127, 129, 132, 135, 138, 140, 143, 146, 148, 151, 154, 157, 159, 162, 165, 167, 170, 173, 175

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=1+sqrt(2) and v=sqrt(2).  Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.
Is this a shifted version of A126281? - R. J. Mathar, Jan 24 2011
The answer to R. J. Mathar's question is no: A126281 contains 65 while this sequence does not. - L. Edson Jeffery, Sep 02 2014

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A182760, A182770.

[Floor(n*(4 + Sqrt(2))/2): n in [1..50]]; // G. C. Greubel, Jan 27 2018

Table[Floor[n*(4 + Sqrt[2])/2], {n, 1, 100}] (* G. C. Greubel, Jan 27 2018 *)

a(n) = floor(n*(4+sqrt(2))/2); \\ Michel Marcus, Sep 02 2014

a(n) = floor(n*(4 + sqrt(2))/2).

cp's OEIS Frontend

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.

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