This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120043 #11 Aug 31 2024 09:59:09
%S A120043 0,0,0,0,0,0,0,0,0,0,0,1,1,5,8,22,47,103,234,492,1082,2271,4867,10349,
%T A120043 21794,45907,96293,202006,421287,879388,1828931,3800227,7882784,
%U A120043 16325796,33771056,69767214,143971956,296771231,611156696,1257374970
%N A120043 Number of 12-almost primes 12ap such that 2^n < 12ap <= 2^(n+1).
%C A120043 The partial sum equals the number of Pi_12(2^n).
%e A120043 (2^12, 2^13] there is one semiprime, namely 6144. 4096 was counted in the previous entry.
%t A120043 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 A120043 t = Table[AlmostPrimePi[12, 2^n], {n, 0, 30}]; Rest@t - Most@t
%o A120043 (Python)
%o A120043 from math import isqrt, prod
%o A120043 from sympy import primerange, integer_nthroot, primepi
%o A120043 def A120043(n):
%o A120043 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 A120043 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 A120043 return -almostprimepi(m:=1<<n,12)+almostprimepi(m<<1,12) # _Chai Wah Wu_, Aug 31 2024
%Y A120043 Cf. A069273, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.
%K A120043 nonn
%O A120043 0,14
%A A120043 _Jonathan Vos Post_ and _Robert G. Wilson v_, Mar 21 2006