A334069 -id:A334069 - cp's OEIS frontend

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

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

nonn

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

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

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

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

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A120035(n):
    x = 1<Chai Wah Wu, Mar 28 2025

A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 11, 10, 7, 2, 1, 18, 22, 13, 7, 2, 1, 31, 42, 30, 14, 7, 2, 1, 54, 82, 60, 34, 15, 7, 2, 1, 97, 157, 125, 71, 36, 15, 7, 2, 1, 172, 304, 256, 152, 77, 37, 15, 7, 2, 1, 309, 589, 513, 325, 168, 81, 37, 15, 7, 2, 1, 564, 1124, 1049, 669, 367, 177, 83, 37

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

			Triangle begins:
  1
  2 1
  4 2 1
  6 6 2 1
  11 10 7 2 1
  18 22 13 7 2 1
  31 42 30 14 7 2 1
  54 82 60 34 15 7 2 1
  97 157 125 71 36 15 7 2 1
  172 304 256 152 77 37 15 7 2 1

Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

Jonathan Vos Post & Robert G. Wilson v, Table of n, a(n) for n = 1..1330
Jonathan Vos Post & Robert G. Wilson v, Regular Triangle of A126279 for the first 52 rows, with some holes.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.

First column: A007053, second column: A125527, third column: A127396, 4th column: A334069. The last row reversed: A052130; the n-th row's sum: A000225 = 2^n -1.

Cf. A126280: same array but for powers of ten.

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 *)
Table[ AlmostPrimePi[m, 2^n], {n, 16}, {m, n}] // Flatten

cp's OEIS Frontend

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

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