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Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.
3

%I #30 Sep 05 2023 22:22:30

%S 0,0,0,1,2,7,14,34,71,152,325,669,1405,2866,5931,12139,24782,50444,

%T 102458,207945,420511,850518,1716168,3460304,6968639,14022029,

%U 28189833,56631732,113697179,228115641,457456902,916899721,1836996851,3678943569,7365141297,14740076678,29490954290

%N Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

%H Robert G. Wilson v, <a href="/A334069/b334069.txt">Table of n, a(n) for n = 1..53</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%F a(n) = A082996(2^n).

%e a(6) = 7 because

%e 16 = 2 * 2 * 2 * 2,

%e 24 = 2 * 2 * 2 * 3,

%e 36 = 2 * 2 * 3 * 3,

%e 40 = 2 * 2 * 2 * 5,

%e 54 = 2 * 3 * 3 * 3,

%e 56 = 2 * 2 * 2 * 7, and

%e 60 = 2 * 2 * 3 * 5

%e are the seven numbers less than 2^6 = 64 that are each the product of four primes.

%t 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]

%Y Cf. A007053, A014613, A082996, A125527, A127396, A126279.

%Y Partial sums of A120035.

%K nonn

%O 1,5

%A _Robert G. Wilson v_, Apr 13 2020