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A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 16
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..100

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.

Main diagonal gives A185232.

Antidiagonal sums give A258652.

Sequence in context: A325201 A260019 A153036 * A258850 A182114 A122950

Adjacent sequences:  A258648 A258649 A258650 * A258652 A258653 A258654

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 06 2015

STATUS

approved

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Last modified December 16 04:05 EST 2019. Contains 330013 sequences. (Running on oeis4.)