A084887 - OEIS

%I #7 Sep 26 2021 09:05:49

%S 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 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 25,100,1,121,121,121,121,121,121,121,121,121,121,1,9,36,4,9,36,4,36

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

%C Arithmetic derivative of k/n = (n*A003415(k)-k*A003415(n))/n^2;

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuotientRule.html">Quotient Rule.</a>

%e ............................. 0

%e ....................... -1/4 ..... 0

%e ................ -1/9 ...... 1/9 ...... 0

%e .... ...... -1/4 ..... -1/4 .... -1/2 ...... 0

%e .. .. -1/25 .... 3/25 ..... 2/25 ..... 16/25 .... 0

%e -5/36 ..... -1/9 ..... -1/4 ..... 1/9 .... -19/36 .... 0.

%t 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 ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;

%t T[n_, k_] := ader[k/n] // Denominator;

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2021 *)

%Y Numerator=A084886, A084885.

%K nonn,tabl

%O 1,2

%A _Reinhard Zumkeller_, Jun 10 2003