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A084885 Triangular array, read by rows: T(n,k) = denominator of arithmetic derivative of n/k, 1<=k<=n.

%I A084885 #8 Feb 16 2025 08:32:49
%S A084885 1,1,1,1,4,1,1,1,9,1,1,4,9,1,1,1,1,1,4,25,1,1,4,9,2,25,36,1,1,1,9,1,
%T A084885 25,9,49,1,1,4,1,4,25,4,49,16,1,1,1,9,4,1,9,49,1,27,1,1,4,9,2,25,36,
%U A084885 49,16,27,100,1,1,1,1,1,25,1,49,4,9,25,121,1
%N A084885 Triangular array, read by rows: T(n,k) = denominator of arithmetic derivative of n/k, 1<=k<=n.
%C A084885 Arithmetic derivative of n/k = (k*A003415(n)-n*A003415(k))/k^2;
%H A084885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuotientRule.html">Quotient Rule.</a>
%e A084885 ......................... 0
%e A084885 ................... 1 ........ 0
%e A084885 ............... 1 .... -1/4 ....... 0
%e A084885 ........... 4 ..... 1 ...... 8/9 ....... 0
%e A084885 ....... 1 ... -3/4 ... -2/9 ...... -1 ...... 0
%e A084885 ... 5 ..... 1 ..... 1 ..... -1/4 .... 19/25 .... 0
%e A084885 1 .. -5/4 ... -4/9 ... -3/2 ... -2/25 ... -29/36 ... 0.
%t A084885 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 A084885 ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;
%t A084885 T[n_, k_] := ader[n/k] // Denominator;
%t A084885 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2021 *)
%Y A084885 Numerator=A084884, A084887.
%K A084885 nonn,tabl
%O A084885 1,5
%A A084885 _Reinhard Zumkeller_, Jun 10 2003