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

%I A258995 #11 May 18 2024 01:54:38
%S A258995 0,5,11,10,4,29,35,41,14,431,599,78,15,38,201,191,25,382,186,43,19,65,
%T A258995 94,3001,535,22,42,633,317,4397,21,141,8,74,574,214,1286,61,253,247,
%U A258995 1417,163,115,217,66,546,138,10631,1997,51,12097,12301,362,26,563,1013
%N A258995 Third pi-based antiderivative of n: the least m such that A258851^3(m) equals n.
%H A258995 Alois P. Heinz, <a href="/A258995/b258995.txt">Table of n, a(n) for n = 0..10000</a>
%F A258995 a(n) = min { m >= 0 : A258851^3(m) = n }.
%F A258995 A258851^3(a(n)) = A258853(a(n)) = n.
%F A258995 a(n) <= A000040^3(n) for n>0.
%F A258995 a(n) <= A258861^3(n).
%p A258995 with(numtheory):
%p A258995 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A258995 a:= proc() local t, a; t, a:= -1, proc() -1 end;
%p A258995       proc(n) local h;
%p A258995         while a(n) = -1 do
%p A258995           t:= t+1; h:= d(d(d(t)));
%p A258995           if a(h) = -1 then a(h):= t fi
%p A258995         od; a(n)
%p A258995       end
%p A258995     end():
%p A258995 seq(a(n), n=0..100);
%t A258995 d[n_] := d[n] = If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#] & /@ FactorInteger[n]]];
%t A258995 A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
%t A258995 a[n_] := A[n, 3];
%t A258995 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, May 17 2024 *)
%Y A258995 Column k=3 of A259016.
%Y A258995 Cf. A000040, A000720, A258851, A258853, A258861, A258862.
%K A258995 nonn
%O A258995 0,2
%A A258995 _Alois P. Heinz_, Jun 16 2015