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A084890 Triangular array, read by rows: T(n,k) = arithmetic derivative of n*k, 1<=k<=n.

%I A084890 #11 Feb 16 2025 08:32:49
%S A084890 0,1,4,1,5,6,4,12,16,32,1,7,8,24,10,5,16,21,44,31,60,1,9,10,32,12,41,
%T A084890 14,12,32,44,80,68,112,92,192,6,21,27,60,39,81,51,156,108,7,24,31,68,
%U A084890 45,92,59,176,123,140,1,13,14,48,16,61,18,140,75,87,22,16,44,60,112,92,156,124,272,216,244,188,384
%N A084890 Triangular array, read by rows: T(n,k) = arithmetic derivative of n*k, 1<=k<=n.
%H A084890 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProductRule.html">Product Rule</a>.
%F A084890 T(n,k) = A003415(n*k) = n*A003415(k)+k*A003415(n), 1<=k<=n.
%F A084890 T(n,1) = A003415(n); n>1.
%F A084890 T(n,2) = A068719(n).
%F A084890 T(n,n) = A068720(n).
%e A084890 ................. 0
%e A084890 .............. 1 ... 4
%e A084890 ........... 1 ... 5 ... 6
%e A084890 ........ 4 .. 12 .. 16 .. 32
%e A084890 ..... 1 ... 7 ... 8 .. 24 .. 10
%e A084890 .. 5 .. 16 .. 21 .. 44 .. 31 .. 60
%e A084890 1 ... 9 .. 10 .. 32 .. 12 .. 41 .. 14.
%t A084890 ader[n_] := ader[n] =Switch[n, 0 | 1, 0, _, If[PrimeQ[n], 1, Sum[Module[{p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
%t A084890 T[n_, k_] := ader[n k];
%t A084890 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 21 2021 *)
%Y A084890 Cf. A084884, A084885, A084886, A084887.
%K A084890 nonn,tabl
%O A084890 1,3
%A A084890 _Reinhard Zumkeller_, Jun 10 2003