A114791 Consider the array T where the element at T(r,c) is the number of n-almost primes less than or equal to 2^m. Sequence read by successive columns beginning at r=c.

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

Offset: 1

Author

Robert G. Wilson v, Feb 21 2006

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

Crossrefs

First row is A007053, Cf. A052130.

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

Formula

Each column sums to 2^n-1.