OFFSET
0,14
COMMENTS
The partial sum equals the number of Pi_12(2^n).
EXAMPLE
(2^12, 2^13] there is one semiprime, namely 6144. 4096 was counted in the previous entry.
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 *)
t = Table[AlmostPrimePi[12, 2^n], {n, 0, 30}]; Rest@t - Most@t
PROG
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120043(n):
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)))
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))
return -almostprimepi(m:=1<<n, 12)+almostprimepi(m<<1, 12) # Chai Wah Wu, Aug 31 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006
STATUS
approved