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

0,8

Links

Table of n, a(n) for n=0..79.

Example

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).

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

Crossrefs

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

The columns are: A057427, A000720.

Sequence in context: A114219 A119339 A037835 * A360099 A106509 A324692

Adjacent sequences: A116430 A116431 A116432 * A116434 A116435 A116436

Keyword

nonn,tabl

Author

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

Extensions

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

Status

approved