A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 3, 1, 1, 0, 0, 4, 2, 1, 0, 0, 0, 4, 2, 1, 1, 0, 0, 0, 7, 2, 1, 1, 0, 0, 0, 0, 7, 4, 1, 1, 0, 0, 0, 0, 0, 8, 4, 2, 1, 0, 0, 0, 0, 0, 0, 8, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 11, 5, 2, 2, 1, 1, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Jul 07 2003

Keywords

• nonn
• tabl

Comments

T(n,1) = A011371(n); T(n,2) = A054861(n) for n>1;

T(n,k) = number of occurrences of prime(k) as factor in numbers <= n (with repetitions);

Sum{T(n,k): 1<=k<=n} = A022559(n);

T(n, A000720(n)) = 1; T(n,k) = 0, A000720(n)

T(n,k) = A115627(n,k) for n > 1 and k=1..A000720(n). - Reinhard Zumkeller, Nov 01 2013

Examples

			0;
1,0;
1,1,0;
3,1,0,0;
3,1,1,0,0;
4,2,1,0,0,0;
4,2,1,1,0,0,0;
7,2,1,1,0,0,0,0;
7,4,1,1,0,0,0,0,0;
8,4,2,1,0,0,0,0,0,0;
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A141809, A115627, A000142.

Programs

Haskell

a085604 n k = a085604_tabl !! (n-2) !! (k-1)
a085604_row 1 = [0]
a085604_row n = a115627_row n ++ (take $ a062298 $ fromIntegral n) [0,0..]
a085604_tabl = map a085604_row [1..]
-- Reinhard Zumkeller, Nov 01 2013

Mathematica

T[n_, k_] := Module[{p = Prime[k], jm}, jm = Floor[Log[p, n]]; Sum[Quotient[n, p^j], {j, 1, jm}]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

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