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

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

Author

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

Keywords

• tabl
• less
• nonn

Examples

			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
		

References

• 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.

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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