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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.

%I A258651 #35 Feb 06 2024 15:31:44
%S A258651 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,
%T A258651 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,
%U A258651 0,0,0,4,0,0,0,32,5,7,11,0,0,0,0,4,0,0,0,80,1,1,1,12
%N A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A258651 Alois P. Heinz, <a href="/A258651/b258651.txt">Antidiagonals n = 0..115, flattened</a>
%H A258651 J. Kovič, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kovic/kovic4.html">The Arithmetic Derivative and Antiderivative</a>, Journal of Integer Sequences 15 (2012), Article 12.3.8
%H A258651 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%F A258651 A(n,k) = A003415^k(n).
%e A258651 Square array A(n,k) begins:
%e A258651   0,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   1,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   2,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   3,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   4,  4,  4,  4,  4,   4,   4,   4,    4,    4, ...
%e A258651   5,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   6,  5,  1,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   7,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
%e A258651   8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
%e A258651   9,  6,  5,  1,  0,   0,   0,   0,    0,    0, ...
%p A258651 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A258651 A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
%p A258651 seq(seq(A(n, h-n), n=0..h), h=0..14);
%t A258651 d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
%t A258651 A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
%t A258651 Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* _Jean-François Alcover_, Apr 27 2017, translated from Maple *)
%Y A258651 Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
%Y A258651 Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.
%Y A258651 Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.
%Y A258651 Main diagonal gives A185232.
%Y A258651 Antidiagonal sums give A258652.
%Y A258651 Cf. also A328383.
%K A258651 nonn,tabl,look
%O A258651 0,6
%A A258651 _Alois P. Heinz_, Jun 06 2015