A114791 Array T where T(n,m), n >= 1, 1 <= m <= n, is the number of n-almost primes less than or equal to 2^m. Array read by columns upwards beginning at n = m.

1, 1, 2, 1, 2, 4, 1, 2, 6, 6, 1, 2, 7, 10, 11, 1, 2, 7, 13, 22, 18, 1, 2, 7, 14, 30, 42, 31, 1, 2, 7, 15, 34, 60, 82, 54, 1, 2, 7, 15, 36, 71, 125, 157, 97, 1, 2, 7, 15, 37, 77, 152, 256, 304, 172, 1, 2, 7, 15, 37, 81, 168, 325, 513, 589, 309, 1, 2, 7, 15, 37, 83, 177, 367, 669, 1049, 1124, 564

Offset

1,3

Comments

Eventually each column reads 1,2,7,15,37,84,187,421,914,... = A052130.

Links

Table of n, a(n) for n=1..78.

Formula

Each column sums to 2^m-1.

Example

Table begins:
1 2 4 6 11 18 31 54  97 172 309  564 1028 1900 3512  6542
  1 2 6 10 22 42 82 157 304 589 1124 2186 4192 8110 15658
    1 2  7 13 30 60 125 256 513 1049 2082 4214 8401 16771
      1  2  7 14 34  71 152 325  669 1405 2866 5931 12139
         1  2  7 15  36  77 168  367  770 1643 3410  7150
            1  2  7  15  37  81  177  392  831 1790  3757
               1  2  7   15  37   83  182  406  867  1880
                  1  2   7   15   37   84  185  414   887
                     1   2    7   15   37   84  186   418

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[n, 2^k], {k, 12}, {n, k, 1, -1}] // Flatten

Crossrefs

Transposition of A092479, omitting A092479 column 0.

First row is A007053.

Cf. A052130.

Sequence in context: A210215 A203647 A376826 * A129994 A208755 A226441

Adjacent sequences: A114788 A114789 A114790 * A114792 A114793 A114794

Keyword

nonn,tabl

Author

Robert G. Wilson v, Feb 21 2006

Extensions

Edited by Peter Munn, Apr 14 2026

Status

approved