A120035
Number of 4-almost primes f such that 2^n < f <= 2^(n+1).
Original entry on oeis.org
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
(2^4, 2^5] there is one semiprime, namely 24. 16 was counted in the previous entry.
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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
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from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A120035(n):
x = 1<Chai Wah Wu, Mar 28 2025
A334069
Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.
Original entry on oeis.org
0, 0, 0, 1, 2, 7, 14, 34, 71, 152, 325, 669, 1405, 2866, 5931, 12139, 24782, 50444, 102458, 207945, 420511, 850518, 1716168, 3460304, 6968639, 14022029, 28189833, 56631732, 113697179, 228115641, 457456902, 916899721, 1836996851, 3678943569, 7365141297, 14740076678, 29490954290
Offset: 1
a(6) = 7 because
16 = 2 * 2 * 2 * 2,
24 = 2 * 2 * 2 * 3,
36 = 2 * 2 * 3 * 3,
40 = 2 * 2 * 2 * 5,
54 = 2 * 3 * 3 * 3,
56 = 2 * 2 * 2 * 7, and
60 = 2 * 2 * 3 * 5
are the seven numbers less than 2^6 = 64 that are each the product of four primes.
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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)]}]; Array[FourAlmostPrimePi[2^#] &, 37]
Showing 1-2 of 2 results.
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