%I #19 Sep 02 2024 13:04:45

%S 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,

%T 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,

%U 125,385,726,788,406,64,1,0,1,5,34,181,669,1688,2613,2249,887,103,1,0,1,5

%N 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.

%e The array begins:

%e 0 0 0 0 0 0 0 0 0 0 0

%e 1 1 1 1 1 1 1 1 1 1 1

%e 1 2 3 5 8 14 23 39 64 103 169

%e 1 2 6 13 34 77 177 406 887 1962 4225

%e 1 3 9 30 90 269 788 2249 6340 17526 47911

%e T(3,2)=3 because there are 3 2-almost primes <= 3^2 = 9, namely 4, 6, and 9 (see A001358).

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

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

%Y The columns are: A057427, A000720.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_ and _Robert G. Wilson v_, Feb 15 2006

%E NAME corrected by _R. J. Mathar_, Jun 20 2021