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Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of n/k, 1<=k<=n.
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%I #7 Sep 26 2021 08:13:47

%S 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 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 1,-9,-8,-5,-6,-49,-4,-31,-19,-67,0,16,5,4,1,68,1,100,-1,8,19,164

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

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

%C T(n,1) = A003415(n); T(n,n) = 0.

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

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

%e ................... 1 ........ 0

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

%e ........... 4 ..... 1 ...... 8/9 ....... 0

%e ....... 1 ... -3/4 ... -2/9 ...... -1 ...... 0

%e ... 5 ..... 1 ..... 1 ..... -1/4 .... 19/25 .... 0

%e 1 .. -5/4 ... -4/9 ... -3/2 ... -2/25 ... -29/36 ... 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[n/k] // Numerator;

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

%Y Denominator=A084885, A084886, A084890.

%K sign,tabl

%O 1,7

%A _Reinhard Zumkeller_, Jun 10 2003