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A084886
Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of k/n, 1<=k<=n.
3
0, -1, 0, -1, 1, 0, -1, -1, -1, 0, -1, 3, 2, 16, 0, -5, -1, -1, 1, -19, 0, -1, 5, 4, 24, 2, 29, 0, -3, -1, -7, -1, -13, -1, -19, 0, -2, -1, -1, 4, -7, 1, -11, 20, 0, -7, -1, -11, 3, -1, 2, -39, 16, -3, 0, -1, 9, 8, 40, 6, 49, 4, 124, 57, 67, 0, -1, -5, -1, -1, -17, -1
OFFSET
1,12
COMMENTS
Arithmetic derivative of k/n = (n*A003415(k)-k*A003415(n))/n^2;
LINKS
Eric Weisstein's World of Mathematics, Quotient Rule.
EXAMPLE
............................. 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.
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] // Numerator;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2021 *)
CROSSREFS
Denominator=A084887, A084884, A084890.
Sequence in context: A346059 A126323 A292123 * A275463 A338280 A304989
KEYWORD
sign,tabl
AUTHOR
Reinhard Zumkeller, Jun 10 2003
STATUS
approved