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.

A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).

Original entry on oeis.org

1, 4, 5, 12, 7, 16, 9, 32, 21, 24, 13, 44, 15, 32, 31, 80, 19, 60, 21, 68, 41, 48, 25, 112, 45, 56, 81, 92, 31, 92, 33, 192, 61, 72, 59, 156, 39, 80, 71, 176, 43, 124, 45, 140, 123, 96, 49, 272, 77, 140, 91, 164, 55, 216, 87, 240, 101, 120
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 26 2002

Keywords

Comments

Terms are either odd or multiples of 4. - Antti Karttunen, Jul 31 2022

Links

Crossrefs

Cf. A003415, A005843, A068720, A068721, A260624, A327862, A327864.
Second diagonal (without the initial 1) in A084890.
Row 1 of A344027.

Programs

  • Magma
    Ad:=func; [Ad(2*n): n in [1..60]]; // Bruno Berselli, Oct 22 2013
  • Mathematica
    ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[2*n]; Array[a, 100] (* Amiram Eldar, Apr 11 2025 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068719(n) = (n + 2*A003415(n)); \\ Antti Karttunen, Jul 31 2022

Formula

a(n) = A003415(A005843(n)).

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

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

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 *)

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 *)
Showing 1-3 of 3 results.

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

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