A356718 T(n,k) is the total number of prime factors, counted with multiplicity, of k!*(n-k)!, for 0 <= k <= n. Triangle read by rows.

0, 0, 0, 1, 0, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 5, 4, 3, 3, 4, 5, 7, 5, 5, 4, 5, 5, 7, 8, 7, 6, 6, 6, 6, 7, 8, 11, 8, 8, 7, 8, 7, 8, 8, 11, 13, 11, 9, 9, 9, 9, 9, 9, 11, 13, 15, 13, 12, 10, 11, 10, 11, 10, 12, 13, 15, 16, 15, 14, 13, 12, 12, 12

Offset

0,7

Comments

k!*(n-k)! is the denominator in binomial(n,k) = n!/(k!*(n-k)!) and all prime factors in the denominator cancel to leave an integer, so that T(n,k) = A022559(n) - A132896(n,k).

Links

Dario T. de Castro, Rows n = 0..140 of triangle, flattened

Formula

T(n,k) = bigomega(k!*(n-k)!), where 0 <= k <= n.

T(n,0) = T(n,n) = A022559(n).

Example

Triangle begins:
  n\k| 0  1  2  3  4  5  6  7
  ---+--------------------------------------
   0 | 0
   1 | 0, 0;
   2 | 1, 0, 1;
   3 | 2, 1, 1, 2;
   4 | 4, 2, 2, 2, 4;
   5 | 5, 4, 3, 3, 4, 5;

Mathematica

T[n_, k_]:=PrimeOmega[Factorial[k]*Factorial[n-k]];
tab=Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]]

Crossrefs

Cf. A007318, A001222, A022559, A132896.

Sequence in context: A322134 A023504 A157905 * A260931 A293819 A027113

Adjacent sequences: A356715 A356716 A356717 * A356719 A356720 A356721

Keyword

nonn,tabl,easy

Author

Dario T. de Castro, Aug 24 2022

Status

approved