cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

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

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 *)

A084886 Triangular array, read by rows: T(n,k) = numerator of arithmetic derivative of k/n, 1<=k<=n.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

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

Crossrefs

Denominator=A084887, A084884, 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[k/n] // Numerator;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2021 *)

A084890 Triangular array, read by rows: T(n,k) = arithmetic derivative of n*k, 1<=k<=n.

Original entry on oeis.org

0, 1, 4, 1, 5, 6, 4, 12, 16, 32, 1, 7, 8, 24, 10, 5, 16, 21, 44, 31, 60, 1, 9, 10, 32, 12, 41, 14, 12, 32, 44, 80, 68, 112, 92, 192, 6, 21, 27, 60, 39, 81, 51, 156, 108, 7, 24, 31, 68, 45, 92, 59, 176, 123, 140, 1, 13, 14, 48, 16, 61, 18, 140, 75, 87, 22, 16, 44, 60, 112, 92, 156, 124, 272, 216, 244, 188, 384
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

Examples

			................. 0
.............. 1 ... 4
........... 1 ... 5 ... 6
........ 4 .. 12 .. 16 .. 32
..... 1 ... 7 ... 8 .. 24 .. 10
.. 5 .. 16 .. 21 .. 44 .. 31 .. 60
1 ... 9 .. 10 .. 32 .. 12 .. 41 .. 14.

Links

Crossrefs

Cf. A084884, A084885, A084886, A084887.

Programs

  • Mathematica
    ader[n_] := 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[n_, k_] := ader[n k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2021 *)

Formula

T(n,k) = A003415(n*k) = n*A003415(k)+k*A003415(n), 1<=k<=n.
T(n,1) = A003415(n); n>1.
T(n,2) = A068719(n).
T(n,n) = A068720(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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