A084884 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of n/k, 1<=k<=n.

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

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• sign
• tabl

Comments

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

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

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

Denominator=A084885, A084886, A084890.

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