A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..115, flattened

J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8

Wikipedia, Arithmetic derivative

Formula

A(n,k) = A003415^k(n).

Example

Square array A(n,k) begins:
  0,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  1,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  2,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  3,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  4,  4,  4,  4,  4,   4,   4,   4,    4,    4, ...
  5,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  6,  5,  1,  0,  0,   0,   0,   0,    0,    0, ...
  7,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
  9,  6,  5,  1,  0,   0,   0,   0,    0,    0, ...

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);

Mathematica

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)

Crossrefs

Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.

Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.

Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.

Main diagonal gives A185232.

Antidiagonal sums give A258652.

Cf. also A328383.

Sequence in context: A325201 A260019 A153036 * A350530 A258850 A182114

Adjacent sequences: A258648 A258649 A258650 * A258652 A258653 A258654

Keyword

nonn,tabl,look

Author

Alois P. Heinz, Jun 06 2015

Status

approved