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A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.

%I A070561 #20 Sep 07 2023 23:40:44
%S A070561 0,11,10,10,9,2,9,8,8,7,1,9,8,8,7,2,7,7,6,4,1,6,6,5,6,3,5,6,3,4,1,4,5,
%T A070561 3,3,2,4,4,4,3,1,5,2,3,2,4,2,3,2,5,1,6,4,9,3,2,5,3,3,2,1,3,3,6,6,3,2,
%U A070561 2,8,6,1,5,5,3,2,2,7,6,4,3,1,5,3,2,8,4,4,4,5,4,1,2,2,2,2,2,2,2,2,2,1,1,1,1
%N A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.
%C A070561 Start with n, repeatedly replace x by x + product of digits of x until the product of digits reaches 0; fecundity = number of steps - 1.
%C A070561 Equivalently, with A230099 = f, a(n) is the number k of distinct values that are obtained with iterations: n, f(n), f(f(n)), f(f(f(n))), ... until a term of this sequence contains a 0. - _Bernard Schott_, Jul 31 2023
%H A070561 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/195-a306-les-nombres-feconds">A306, Les nombres féconds</a> (in French).
%F A070561 a(n) = 1 iff n positive is in A011540. - _Bernard Schott_, Jul 31 2023
%e A070561 1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 ->102 -> 102 -> ... has fecundity 10.
%t A070561 f[n_] := Length@FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 1; f[0] = 0; Array[f, 105, 0] (* _Robert G. Wilson v_, Jun 27 2010 *)
%Y A070561 Cf. A011540, A070560, A070562, A031346, A003001, A230099.
%K A070561 nonn,easy,base
%O A070561 0,2
%A A070561 _N. J. A. Sloane_, May 07 2002
%E A070561 More terms from _Robert G. Wilson v_, Jun 27 2010