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A084884 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of n/k, 1<=k<=n.

%I A084884 #8 Feb 16 2025 08:32:49
%S A084884 0,1,0,1,-1,0,4,1,8,0,1,-3,-2,-1,0,5,1,1,-1,19,0,1,-5,-4,-3,-2,-29,0,
%T A084884 12,4,28,1,52,8,76,0,6,3,1,-3,21,-1,33,-15,0,7,1,11,-3,1,-2,39,-1,1,0,
%U A084884 1,-9,-8,-5,-6,-49,-4,-31,-19,-67,0,16,5,4,1,68,1,100,-1,8,19,164
%N A084884 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of n/k, 1<=k<=n.
%C A084884 Arithmetic derivative of n/k = (k*A003415(n)-n*A003415(k))/k^2;
%C A084884 T(n,1) = A003415(n); T(n,n) = 0.
%H A084884 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuotientRule.html">Quotient Rule.</a>
%e A084884 ......................... 0
%e A084884 ................... 1 ........ 0
%e A084884 ............... 1 .... -1/4 ....... 0
%e A084884 ........... 4 ..... 1 ...... 8/9 ....... 0
%e A084884 ....... 1 ... -3/4 ... -2/9 ...... -1 ...... 0
%e A084884 ... 5 ..... 1 ..... 1 ..... -1/4 .... 19/25 .... 0
%e A084884 1 .. -5/4 ... -4/9 ... -3/2 ... -2/25 ... -29/36 ... 0.
%t A084884 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 A084884 ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;
%t A084884 T[n_, k_] := ader[n/k] // Numerator;
%t A084884 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2021 *)
%Y A084884 Denominator=A084885, A084886, A084890.
%K A084884 sign,tabl
%O A084884 1,7
%A A084884 _Reinhard Zumkeller_, Jun 10 2003