cp's OEIS Frontend

A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).

%I A120041 #23 Sep 01 2024 10:29:14
%S A120041 0,0,0,0,0,0,0,0,0,1,1,5,8,22,47,103,233,487,1072,2246,4803,10202,
%T A120041 21440,45115,94434,197891,412010,858846,1783610,3700698,7665755,
%U A120041 15853990,32750248,67564405,139238488,286625278,589472979,1211146741,2486322304
%N A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).
%H A120041 Chai Wah Wu, <a href="/A120041/b120041.txt">Table of n, a(n) for n = 0..47</a>
%F A120041 a(n) ~ 2^n log^9 n/(725760 n log 2). [_Charles R Greathouse IV_, Dec 28 2011]
%t A120041 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 A120041 t = Table[AlmostPrimePi[10, 2^n], {n, 0, 39}]; Rest@t - Most@t
%o A120041 (Python)
%o A120041 from math import isqrt, prod
%o A120041 from sympy import primerange, integer_nthroot, primepi
%o A120041 def A120041(n):
%o A120041     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 A120041     def almostprimepi(n,k): return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
%o A120041     return -almostprimepi(m:=1<<n,10)+almostprimepi(m<<1,10) # _Chai Wah Wu_, Aug 31 2024
%Y A120041 Cf. A046314, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.
%K A120041 nonn
%O A120041 0,12
%A A120041 _Jonathan Vos Post_ and _Robert G. Wilson v_, Mar 21 2006