A258862 Second pi-based antiderivative of n: the least m such that A258851^2(m) equals n.

0, 3, 5, 11, 4, 10, 41, 13, 15, 83, 109, 29, 19, 35, 191, 43, 30, 277, 74, 14, 8, 42, 77, 431, 461, 21, 22, 563, 66, 599, 26, 78, 12, 61, 141, 163, 877, 18, 214, 218, 226, 38, 114, 201, 105, 1201, 215, 1297, 302, 55, 1447, 1471, 89, 25, 103, 170, 58, 291, 51

Offset: 0

Author

Alois P. Heinz, Jun 12 2015

Keywords

• nonn

Links

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

Crossrefs

Column k=2 of A259016.

Cf. A000040, A000720, A258851, A258852, A258861, A258995.

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:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(t));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

Mathematica

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, 2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

Formula

a(n) = min { m >= 0 : A258851^2(m) = n }.

A258851^2(a(n)) = A258852(a(n)) = n.

a(n) <= A000040^2(n) for n>0.

a(n) <= A258861^2(n); a(21) = 42 < A258861^2(21) = A258861(22) = 79; A258851^2(42) = A258851^2(79) = 21.