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

0,6

Links

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

Formula

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

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

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

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,  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, ...

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 *)

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.

Sequence in context: A153036 A258651 A350530 * A182114 A122950 A116489

Adjacent sequences: A258847 A258848 A258849 * A258851 A258852 A258853

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 12 2015

Status

approved