A092479 T(n,k) is the number of numbers <= 2^n having exactly k prime factors (with repetitions), 0<=k<=n, triangle read by rows.

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

Offset

0,5

Comments

As n, respectively k, increases, the reversed rows, respectively columns, approach the same limit, given by A052130. - Peter Munn, Mar 26 2026

References

Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.

Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..377 (rows n = 0..26, flattened)

Index entries for sequences related to numbers of primes in various ranges

Formula

T(n,0) = 1; T(n,1) = A007053(n,1) for n>0; T(n,n) = 1.

Sum_{k=0..n} T(n,k) = 2^n.

Example

Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  2,  1;
  1,  6,  6,  2,  1;
  1, 11, 10,  7,  2,  1;
  1, 18, 22, 13,  7,  2,  1;
  1, 31, 42, 30, 14,  7,  2,  1;
  1, 54, 82, 60, 34, 15,  7,  2,  1;
  ...

Mathematica

row[n_] := Module[{i, v = Table[0, {n}]}, For[i = 2, i <= 2^n, i++, v[[PrimeOmega[i]]]++]; Prepend[v, 1]];
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 23 2021, after PARI code *)

Prog

(PARI) row(n) = {v = vector(n); for (i=2, 2^n, v[bigomega(i)]++; ); concat(1, v); }
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Aug 11 2015

Crossrefs

Cf. A000079 (row sums), A001222, A052130.

Columns: A000012 (0), A007053 (1), A125527 (2), A127396 (3), A334069 (4).

A126279 is a subtable (omits column 0).

Sequence in context: A089688 A229706 A319421 * A124022 A098063 A209438

Adjacent sequences: A092476 A092477 A092478 * A092480 A092481 A092482

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Mar 27 2004

Status

approved