A052130 - OEIS

%I #56 Dec 04 2024 05:49:57

%S 1,2,7,15,37,84,187,421,914,2001,4283,9184,19611,41604,87993,185387,

%T 389954,817053,1709640,3567978,7433670,15460810,32103728,66567488,

%U 137840687,285076323,588891185,1215204568,2505088087,5159284087

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

%C 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

%C 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

%C From _Robert G. Wilson v_, Apr 13 2020: (Start)

%C This sequence shows a sufficiently large row of A126279 read backwards or a sufficiently large column of A126279 read vertically.

%C 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)

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

%H Martin Raab, <a href="/A052130/b052130.txt">Table of n, a(n) for n = 0..41</a> (Terms 0..35 from Robert G. Wilson v)

%H <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>

%e Between 1 and 2^m there is just one number with m prime factors, namely 2^m, so a(0) = 1.

%e 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.

%t 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 *)

%t 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 *)

%o (Python)

%o from math import prod, isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A052130(n):

%o     if n<=1: return n+1

%o     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)))

%o     k = 1

%o     while 3**k<(r:=1<<n+k): k+=1

%o     return int(sum(primepi(r//prod(c[1] for c in a))-a[-1][0] for a in g(r,0,1,1,k))) # _Chai Wah Wu_, Dec 03 2024

%Y Cf. A117630, A126279, A126281.

%K nonn,nice

%O 0,2

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

%E More terms from _David W. Wilson_, Feb 01 2000

%E a(24)-a(29) from _Robert G. Wilson v_, Feb 21 2006