A132896 -id:A132896 - cp's OEIS frontend

A048571 Triangle read by rows: T(n,k) = number of distinct prime factors of C(n,k).

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 1, 2, 2, 3, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, 0, 2, 2, 3, 4, 3, 4, 3, 2, 2, 0, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, 0, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 2, 0

Offset: 0

N. J. A. Sloane

			Triangle begins:
0
0,0
0,1,0
0,1,1,0
0,1,2,1,0
0,1,2,2,1,0
0,2,2,2,2,2,0
0,1,2,2,2,2,1,0
...

T. D. Noe, Rows n=0..100 of triangle, flattened
Pierre Goetgheluck, On prime divisors of binomial coefficients, Math. Comp. 51 (1988), no. 183, 325-329.

Cf. A048273, A132896.

Cf. A001221, A007318.

Flatten[Table[b=Binomial[n,k]; If[b==1, 0, Length[FactorInteger[b]]], {n,0,12}, {k,0,n}]] (* T. D. Noe, Oct 19 2007, Apr 03 2012 *)
Table[PrimeNu[Binomial[n,k]],{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 11 2019 *)

T(n, k) = A001221(A007318(n, k)). - Michel Marcus, Nov 04 2020

Edited Oct 06 2007 at the suggestion of T. D. Noe

Corrected by T. D. Noe, Oct 19 2007

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.

Original entry on oeis.org

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

Dario T. de Castro, Aug 24 2022

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).

			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;

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

Cf. A007318, A001222, A022559, A132896, A303279.

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

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

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

cp's OEIS Frontend

A048571 Triangle read by rows: T(n,k) = number of distinct prime factors of C(n,k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions