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%I #29 Apr 30 2020 10:16:07
%S 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,
%T 82,60,34,15,7,2,1,97,157,125,71,36,15,7,2,1,172,304,256,152,77,37,15,
%U 7,2,1,309,589,513,325,168,81,37,15,7,2,1,564,1124,1049,669,367,177,83,37
%N Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.
%D Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.
%D Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.
%H Jonathan Vos Post & Robert G. Wilson v, <a href="/A126279/b126279.txt">Table of n, a(n) for n = 1..1330</a>
%H Jonathan Vos Post & Robert G. Wilson v, <a href="/A126279/a126279.txt">Regular Triangle of A126279 for the first 52 rows, with some holes.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%e Triangle begins:
%e 1
%e 2 1
%e 4 2 1
%e 6 6 2 1
%e 11 10 7 2 1
%e 18 22 13 7 2 1
%e 31 42 30 14 7 2 1
%e 54 82 60 34 15 7 2 1
%e 97 157 125 71 36 15 7 2 1
%e 172 304 256 152 77 37 15 7 2 1
%t 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[m, 2^n], {n, 16}, {m, n}] // Flatten
%Y 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.
%Y Cf. A126280: same array but for powers of ten.
%K tabl,less,nonn
%O 1,2
%A _Jonathan Vos Post_ & _Robert G. Wilson v_, Dec 22 2006
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