A120035 Number of 4-almost primes f such that 2^n < f <= 2^(n+1).

0, 0, 0, 1, 1, 5, 7, 20, 37, 81, 173, 344, 736, 1461, 3065, 6208, 12643, 25662, 52014, 105487, 212566, 430007, 865650, 1744136, 3508335, 7053390, 14167804, 28441899, 57065447, 114418462, 229341261, 459442819, 920097130, 1841946718, 3686197728

Offset

0,6

Comments

The partial sum equals the number of Pi_4(2^n) = A334069(n).

Links

Table of n, a(n) for n=0..34.

Example

(2^4, 2^5] there is one semiprime, namely 24. 16 was counted in the previous entry.

Mathematica

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; t = Table[ FourAlmostPrimePi[2^n], {n, 0, 37}]; Rest@t - Most@t

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A120035(n):
    x = 1<<n
    y = x<<1
    return sum(primepi(y//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(y, 4)[0]+1)) for b, m in enumerate(primerange(k, integer_nthroot(y//k, 3)[0]+1), a) for c, r in enumerate(primerange(m, isqrt(y//(k*m))+1), b))-sum(primepi(x//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(x, 4)[0]+1)) for b, m in enumerate(primerange(k, integer_nthroot(x//k, 3)[0]+1), a) for c, r in enumerate(primerange(m, isqrt(x//(k*m))+1), b)) # Chai Wah Wu, Mar 28 2025

Crossrefs

Cf. A014613, A082996, A114106, A036378, A120033 - A120043, A334069.

Sequence in context: A249047 A258282 A192422 * A198302 A091154 A384347

Adjacent sequences: A120032 A120033 A120034 * A120036 A120037 A120038

Keyword

nonn

Author

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

Status

approved