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A084886 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of k/n, 1<=k<=n.

%I A084886 #8 Feb 16 2025 08:32:49
%S A084886 0,-1,0,-1,1,0,-1,-1,-1,0,-1,3,2,16,0,-5,-1,-1,1,-19,0,-1,5,4,24,2,29,
%T A084886 0,-3,-1,-7,-1,-13,-1,-19,0,-2,-1,-1,4,-7,1,-11,20,0,-7,-1,-11,3,-1,2,
%U A084886 -39,16,-3,0,-1,9,8,40,6,49,4,124,57,67,0,-1,-5,-1,-1,-17,-1
%N A084886 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of k/n, 1<=k<=n.
%C A084886 Arithmetic derivative of k/n = (n*A003415(k)-k*A003415(n))/n^2;
%H A084886 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuotientRule.html">Quotient Rule.</a>
%e A084886 ............................. 0
%e A084886 ....................... -1/4 ..... 0
%e A084886 ................ -1/9 ...... 1/9 ...... 0
%e A084886 .... ...... -1/4 ..... -1/4 .... -1/2 ...... 0
%e A084886 .. .. -1/25 .... 3/25 ..... 2/25 ..... 16/25 .... 0
%e A084886 -5/36 ..... -1/9 ..... -1/4 ..... 1/9 .... -19/36 .... 0.
%t A084886 ader[n_Integer] := 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 A084886 ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;
%t A084886 T[n_, k_] := ader[k/n] // Numerator;
%t A084886 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2021 *)
%Y A084886 Denominator=A084887, A084884, A084890.
%K A084886 sign,tabl
%O A084886 1,12
%A A084886 _Reinhard Zumkeller_, Jun 10 2003