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A342718 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> A000203(x)/A000005(x) to reach a number that is not an integer, when starting from x = n.

%I A342718 #29 Mar 23 2021 15:56:43
%S A342718 0,1,2,1,3,3,2,1,1,1,4,1,3,4,4,1,2,1,2,3,2,2,2,1,1,1,2,1,5,2,2,1,2,1,
%T A342718 2,1,3,5,5,1,3,2,3,5,4,2,2,1,3,1,2,1,3,5,2,5,4,1,3,5,3,2,1,1,3,2,2,3,
%U A342718 2,2,2,1,4,1,1,1,2,3,2,1,1,1
%N A342718 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> A000203(x)/A000005(x) to reach a number that is not an integer, when starting from x = n.
%C A342718 a(n) = 1 for n >= 2, n from A049642.
%e A342718 n = 3; 3 -> s(3)/d(3) = 2 -> s(2)/d(2) = 3/2, a(3) = 2;
%e A342718 n = 11; 11 -> s(11)/d(11) = 6 -> s(6)/d(6) = 3 -> s(3)/d(3) = 2 -> s(2)/d(2) = 3/2, a(11) = 4;
%e A342718 n = 20; 20 -> s(20)/d(20) = 7 -> s(7)/d(7) = 4 -> s(4)/d(4) =  7/3, a(20) = 3;
%e A342718 s(x) = A000203(x), d(x) = A000005(x).
%t A342718 f[n_] := Divide @@ DivisorSigma[{1, 0}, n]; a[n_] := Length @ NestWhileList[f, n, IntegerQ[#] && # > 1 &] - 1; Array[a, 100] (* _Amiram Eldar_, Mar 19 2021 *)
%o A342718 (PARI) a(n) = if (n==1, 0, my(nb=1, k); while(denominator(k=sigma(n)/numdiv(n)) == 1, n = k; nb++); nb); \\ _Michel Marcus_, Mar 21 2021
%Y A342718 Cf. A000005, A000203, A003601, A049642.
%Y A342718 Cf. A019294 for a similar iteration.
%K A342718 nonn
%O A342718 1,3
%A A342718 _Ctibor O. Zizka_, Mar 19 2021