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

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, 16, 4, 16, 4, 16, 2, 16, 1, 27, 27, 9, 27, 27, 9, 27, 27, 1, 100, 25, 100, 25, 4, 25, 100, 25, 100, 1, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 1, 9, 36, 4, 9, 36, 4, 36

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• nonn
• tabl

Comments

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

Examples

			............................. 0
....................... -1/4 ..... 0
................ -1/9 ...... 1/9 ...... 0
.... ...... -1/4 ..... -1/4 .... -1/2 ...... 0
.. .. -1/25 .... 3/25 ..... 2/25 ..... 16/25 .... 0
-5/36 ..... -1/9 ..... -1/4 ..... 1/9 .... -19/36 .... 0.
		

Links

• Eric Weisstein's World of Mathematics, Quotient Rule.

Crossrefs

Numerator=A084886, A084885.

Programs

Mathematica

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]}]]];
ader[Rational[n_, k_]] := (ader[n] k - ader[k] n)/k^2;
T[n_, k_] := ader[k/n] // Denominator;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2021 *)