A258850 A(n,k) = k-th pi-based 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, 2, 4, 0, 0, 0, 1, 4, 5, 0, 0, 0, 0, 4, 3, 6, 0, 0, 0, 0, 4, 2, 7, 7, 0, 0, 0, 0, 4, 1, 4, 4, 8, 0, 0, 0, 0, 4, 0, 4, 4, 12, 9, 0, 0, 0, 0, 4, 0, 4, 4, 20, 12, 10, 0, 0, 0, 0, 4, 0, 4, 4, 32, 20, 11, 11, 0, 0, 0, 0, 4, 0, 4, 4, 80, 32, 5, 5, 12

Offset: 0

Author

Alois P. Heinz, Jun 12 2015

Keywords

• nonn
• tabl

Examples

			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,  2,  1,  0,  0,   0,   0,    0,     0,     0, ...
  4,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  5,  3,  2,  1,  0,   0,   0,    0,     0,     0, ...
  6,  7,  4,  4,  4,   4,   4,    4,     4,     4, ...
  7,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
  9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..31, flattened

Crossrefs

Columns k=0-10 give: A001477, A258851, A258852, A258853, A258854, A258855, A258856, A258857, A258858, A258859, A258860.

Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.

Antidiagonal sums give A258847.

Main diagonal gives A258849.

Cf. A000720, A258975, A259016.

Programs

Maple

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/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*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* Jean-François Alcover, Apr 24 2016, adapted from Maple *)

Formula

A(n,k) = A258851^k(n).

A(A259016(n,k),k) = n.

A(A258975(n),n) = 1.