A259016 A(n,k) = k-th pi-based antiderivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 2, 2, 0, 3, 3, 3, 0, 5, 5, 5, 4, 0, 11, 11, 11, 4, 5, 0, 10, 10, 10, 4, 11, 6, 0, 29, 29, 29, 4, 10, 13, 7, 0, 78, 78, 78, 4, 29, 41, 6, 8, 0, 141, 141, 141, 4, 78, 35, 13, 19, 9, 0, 266, 266, 266, 4, 141, 38, 41, 15, 23, 10, 0, 147, 147, 147, 4, 266, 163, 35, 14, 83, 29, 11

Offset: 0

Author

Alois P. Heinz, Jun 16 2015

Keywords

• nonn
• tabl

Examples

			A(5,3) = 29 -> 10 -> 11 -> 5.
A(5,4) = 78 -> 127 -> 31 -> 11 -> 5.
Square array A(n,k) begins:
  0,  0,  0,   0,    0,     0,     0,      0,     0,      0, ...
  1,  2,  3,   5,   11,    10,    29,     78,   141,    266, ...
  2,  3,  5,  11,   10,    29,    78,    141,   266,    147, ...
  3,  5, 11,  10,   29,    78,   141,    266,   147,    194, ...
  4,  4,  4,   4,    4,     4,     4,      4,     4,      4, ...
  5, 11, 10,  29,   78,   141,   266,    147,   194,   1181, ...
  6, 13, 41,  35,   38,   163,   138,    253,   346,   1383, ...
  7,  6, 13,  41,   35,    38,   163,    138,   253,    346, ...
  8, 19, 15,  14,   43,   191,   201,    217,  1113,   1239, ...
  9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..20, flattened

Crossrefs

Columns k=0-3 give: A001477, A258861, A258862, A258995.

Rows n=0,1,4,7,8,9 give: A000004, A258975, A010709, A259168, A259169, A259170.

Cf. A000040, A000720, A258850, A258851.

Programs

Maple

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc() local t, A; t, A:= proc()-1 end, proc()-1 end;
      proc(n, k) local h;
        while A(n, k) = -1 do
          t(k):= t(k)+1; h:= (d@@k)(t(k));
          if A(h, k) = -1 then A(h, k):= t(k) fi
        od; A(n, k)
      end
    end():
seq(seq(A(n, h-n), n=0..h), h=0..12);

Mathematica

d[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
Table[A[n, k-n], {k, 0, 12}, {n, 0, k}] // Flatten (* Jean-François Alcover, Mar 20 2017 *)

Formula

A(n,k) = min { m >= 0 : A258851^k(m) = n }.

A258850(A(n,k),k) = n.

A(n,k) <= A000040^k(n) for n>0.