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

0,5

Links

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

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.

Example

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, ...

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 *)

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.

Sequence in context: A288530 A295675 A135356 * A216504 A216673 A363532

Adjacent sequences: A259013 A259014 A259015 * A259017 A259018 A259019

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 16 2015

Status

approved