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

%I A258862 #21 May 18 2024 01:54:45
%S A258862 0,3,5,11,4,10,41,13,15,83,109,29,19,35,191,43,30,277,74,14,8,42,77,
%T A258862 431,461,21,22,563,66,599,26,78,12,61,141,163,877,18,214,218,226,38,
%U A258862 114,201,105,1201,215,1297,302,55,1447,1471,89,25,103,170,58,291,51
%N A258862 Second pi-based antiderivative of n: the least m such that A258851^2(m) equals n.
%H A258862 Alois P. Heinz, <a href="/A258862/b258862.txt">Table of n, a(n) for n = 0..10000</a>
%F A258862 a(n) = min { m >= 0 : A258851^2(m) = n }.
%F A258862 A258851^2(a(n)) = A258852(a(n)) = n.
%F A258862 a(n) <= A000040^2(n) for n>0.
%F A258862 a(n) <= A258861^2(n); a(21) = 42 < A258861^2(21) = A258861(22) = 79; A258851^2(42) = A258851^2(79) = 21.
%p A258862 with(numtheory):
%p A258862 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A258862 a:= proc() local t, a; t, a:= -1, proc() -1 end;
%p A258862       proc(n) local h;
%p A258862         while a(n) = -1 do
%p A258862           t:= t+1; h:= d(d(t));
%p A258862           if a(h) = -1 then a(h):= t fi
%p A258862         od; a(n)
%p A258862       end
%p A258862     end():
%p A258862 seq(a(n), n=0..100);
%t A258862 d[n_] := d[n] = If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
%t A258862 A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
%t A258862 a[n_] := A[n, 2];
%t A258862 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, May 17 2024 *)
%Y A258862 Column k=2 of A259016.
%Y A258862 Cf. A000040, A000720, A258851, A258852, A258861, A258995.
%K A258862 nonn
%O A258862 0,2
%A A258862 _Alois P. Heinz_, Jun 12 2015