A197871 Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

0, 2, 1, 4, 3, 1, 6, 6, 2, 1, 9, 9, 4, 2, 11, 13, 7, 3, 1, 15, 17, 10, 4, 2, 18, 22, 13, 7, 2, 1, 22, 26, 19, 8, 4, 1, 25, 34, 22, 12, 4, 2, 30, 40, 28, 13, 7, 2, 34, 48, 32, 18, 7, 3, 1, 39, 56, 38, 21, 9, 4, 1, 44, 62, 48, 24, 11, 4, 2, 48, 75, 51, 29, 13, 6, 2

Offset

1,2

References

G. J. O. Jameson, The Prime Number Theorem, Cambridge, 2004, p.145.

Links

Table of n, a(n) for n=1..76.

Example

In the third row, reading from the left, 6 is the number of primes <= 16, 6 is the number of semiprimes <= 16, 2 is the number of numbers with three prime divisors (repetitions allowed) <= 16, and 1 is the number of numbers with four divisors <= 16.
The triangle begins:
   0
   2  1
   4  3  1
   6  6  2  1
   9  9  4  2
  11 13  7  3  1
  15 17 10  4  2
  ...

Mathematica

Join[{0}, Flatten[Table[Transpose[Tally[Table[Plus @@ Last /@ FactorInteger[i], {i, 2, n^2}]]][[2]], {n, 2, 15}]]]

Prog

(PARI) T(n, k) = #select(x->(bigomega(x) == k), [1..n^2]);
row(n) = my(v = vector(n, k, T(n, k))); my(pos); for (k=1, n, if (v[k], pos=k)); Vec(v, pos); \\ Michel Marcus, Aug 16 2022

Crossrefs

Similar to A052130.

Sequence in context: A187883 A134543 A305540 * A093010 A179000 A210559

Adjacent sequences: A197868 A197869 A197870 * A197872 A197873 A197874

Keyword

nonn,tabf

Author

Daniel Tisdale, Oct 18 2011

Status

approved