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

Author

Reinhard Zumkeller, Mar 27 2004

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A001222.

Programs

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 *)

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

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.