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

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, 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, 49, 16, 27, 100, 1, 1, 1, 1, 1, 25, 1, 49, 4, 9, 25, 121, 1

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• nonn
• tabl

Comments

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

Examples

			......................... 0
................... 1 ........ 0
............... 1 .... -1/4 ....... 0
........... 4 ..... 1 ...... 8/9 ....... 0
....... 1 ... -3/4 ... -2/9 ...... -1 ...... 0
... 5 ..... 1 ..... 1 ..... -1/4 .... 19/25 .... 0
1 .. -5/4 ... -4/9 ... -3/2 ... -2/25 ... -29/36 ... 0.
		

Links

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

Crossrefs

Numerator=A084884, A084887.

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[n/k] // Denominator;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2021 *)