A258853 -id:A258853 - cp's OEIS frontend

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

Alois P. Heinz, Jun 12 2015

			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, ...

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

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.

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);

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 *)

A(n,k) = A258851^k(n).
A(A259016(n,k),k) = n.
A(A258975(n),n) = 1.

A258995 Third pi-based antiderivative of n: the least m such that A258851^3(m) equals n.

Original entry on oeis.org

0, 5, 11, 10, 4, 29, 35, 41, 14, 431, 599, 78, 15, 38, 201, 191, 25, 382, 186, 43, 19, 65, 94, 3001, 535, 22, 42, 633, 317, 4397, 21, 141, 8, 74, 574, 214, 1286, 61, 253, 247, 1417, 163, 115, 217, 66, 546, 138, 10631, 1997, 51, 12097, 12301, 362, 26, 563, 1013

Offset: 0

Alois P. Heinz, Jun 16 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=3 of A259016.

Cf. A000040, A000720, A258851, A258853, A258861, A258862.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(d(t)));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

d[n_] := 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]]];
a[n_] := A[n, 3];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

a(n) = min { m >= 0 : A258851^3(m) = n }.
A258851^3(a(n)) = A258853(a(n)) = n.
a(n) <= A000040^3(n) for n>0.
a(n) <= A258861^3(n).

cp's OEIS Frontend

A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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