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

Links

Table of n, a(n) for n=0..90.

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;
  ...

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. A001222.

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