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.
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.
AUTHOR
Alois P. Heinz, Jun 06 2015
STATUS
approved