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A084887 Triangular array, read by rows: T(n,k) = denominator of arithmetic derivative of k/n, 1<=k<=n.

%I A084887 #8 Feb 16 2025 08:32:49
%S A084887 1,4,1,9,9,1,4,4,2,1,25,25,25,25,1,36,9,4,9,36,1,49,49,49,49,49,49,1,
%T A084887 16,4,16,4,16,2,16,1,27,27,9,27,27,9,27,27,1,100,25,100,25,4,25,100,
%U A084887 25,100,1,121,121,121,121,121,121,121,121,121,121,1,9,36,4,9,36,4,36
%N A084887 Triangular array, read by rows: T(n,k) = denominator of arithmetic derivative of k/n, 1<=k<=n.
%C A084887 Arithmetic derivative of k/n = (n*A003415(k)-k*A003415(n))/n^2;
%H A084887 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuotientRule.html">Quotient Rule.</a>
%e A084887 ............................. 0
%e A084887 ....................... -1/4 ..... 0
%e A084887 ................ -1/9 ...... 1/9 ...... 0
%e A084887 .... ...... -1/4 ..... -1/4 .... -1/2 ...... 0
%e A084887 .. .. -1/25 .... 3/25 ..... 2/25 ..... 16/25 .... 0
%e A084887 -5/36 ..... -1/9 ..... -1/4 ..... 1/9 .... -19/36 .... 0.
%t A084887 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 A084887 ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;
%t A084887 T[n_, k_] := ader[k/n] // Denominator;
%t A084887 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2021 *)
%Y A084887 Numerator=A084886, A084885.
%K A084887 nonn,tabl
%O A084887 1,2
%A A084887 _Reinhard Zumkeller_, Jun 10 2003