A116433 Consider the array T(r,c) where is the number of c-almost primes less than or equal to r^c, r >= 1, c >= 0. Read the array by antidiagonals.

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 3, 6, 5, 1, 0, 1, 3, 9, 13, 8, 1, 0, 1, 4, 13, 30, 34, 14, 1, 0, 1, 4, 17, 50, 90, 77, 23, 1, 0, 1, 4, 22, 82, 200, 269, 177, 39, 1, 0, 1, 4, 26, 125, 385, 726, 788, 406, 64, 1, 0, 1, 5, 34, 181, 669, 1688, 2613, 2249, 887, 103, 1, 0, 1, 5

Offset: 0

Author

Paul D. Hanna and Robert G. Wilson v, Feb 15 2006

Keywords

• nonn
• tabl

Examples

			The array begins:
  0 0 0 0 0 0 0 0 0 0 0
  1 1 1 1 1 1 1 1 1 1 1
  1 2 3 5 8 14 23 39 64 103 169
  1 2 6 13 34 77 177 406 887 1962 4225
  1 3 9 30 90 269 788 2249 6340 17526 47911
T(3,2)=3 because there are 3 2-almost primes <= 3^2 = 9, namely 4, 6, and 9 (see A001358).
		

Crossrefs

Cf. The rows are: A000004, A000012, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

The columns are: A057427, A000720.

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[ If[k == 0, 1, AlmostPrimePi[n - k + 1, k^(n - k + 1)]], {n, 0, 7}, {k, n, 0, -1}] // Flatten

Extensions

NAME corrected by R. J. Mathar, Jun 20 2021