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

%I A085604 #14 Sep 19 2021 10:47:02
%S A085604 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,
%T A085604 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,
%U A085604 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
%N A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.
%C A085604 T(n,1) = A011371(n); T(n,2) = A054861(n) for n>1;
%C A085604 T(n,k) = number of occurrences of prime(k) as factor in numbers <= n (with repetitions);
%C A085604 Sum{T(n,k): 1<=k<=n} = A022559(n);
%C A085604 T(n, A000720(n)) = 1; T(n,k) = 0, A000720(n)<k<n.
%C A085604 T(n,k) = A115627(n,k) for n > 1 and k=1..A000720(n). - _Reinhard Zumkeller_, Nov 01 2013
%H A085604 Reinhard Zumkeller, <a href="/A085604/b085604.txt">Rows n = 1..125 of triangle, flattened</a>
%e A085604 0;
%e A085604 1,0;
%e A085604 1,1,0;
%e A085604 3,1,0,0;
%e A085604 3,1,1,0,0;
%e A085604 4,2,1,0,0,0;
%e A085604 4,2,1,1,0,0,0;
%e A085604 7,2,1,1,0,0,0,0;
%e A085604 7,4,1,1,0,0,0,0,0;
%e A085604 8,4,2,1,0,0,0,0,0,0;
%t A085604 T[n_, k_] := Module[{p = Prime[k], jm}, jm = Floor[Log[p, n]]; Sum[Quotient[n, p^j], {j, 1, jm}]];
%t A085604 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2021 *)
%o A085604 (Haskell)
%o A085604 a085604 n k = a085604_tabl !! (n-2) !! (k-1)
%o A085604 a085604_row 1 = [0]
%o A085604 a085604_row n = a115627_row n ++ (take $ a062298 $ fromIntegral n) [0,0..]
%o A085604 a085604_tabl = map a085604_row [1..]
%o A085604 -- _Reinhard Zumkeller_, Nov 01 2013
%Y A085604 Cf. A141809, A115627, A000142.
%K A085604 nonn,tabl
%O A085604 1,7
%A A085604 _Reinhard Zumkeller_, Jul 07 2003