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A099646 Function f(n) = 1 + Sum(digit^2 of n) is iterated and a(n) is the length of terminal cycle at initial value n.

%I A099646 #20 Jun 29 2025 01:54:10
%S A099646 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%T A099646 1,1,9,9,9,9,9,9,9,9,9,1,9,9,9,9,9,9,1,9,9,9,1,9,9,9,9,9,1,1,9,9,9,9,
%U A099646 9,9,9,9,9,9,1,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A099646 Function f(n) = 1 + Sum(digit^2 of n) is iterated and a(n) is the length of terminal cycle at initial value n.
%C A099646 Iteration g(x) applied in A031176 is slightly modified to obtain actual function: f(x) = 1 + g(x). Cases of a(n) = 1 (n = 35, 36, 46, 53, 57, 63, 64, 75, 135, ...) are analogous to happy numbers A007770.
%H A099646 Michael De Vlieger, <a href="/A099646/b099646.txt">Table of n, a(n) for n = 1..10000</a>
%e A099646 For n = 1: iteration-list= {1,2,5,26,41,18,66,73,59,107,51,[27,54,42,21,6,37,59,107,51],27...
%e A099646 with t = 11 transient and c = a(1) = 9, the cycle-length;
%e A099646 For n = 35: list={36,46,53,[35],35,...} with transient t = 3, c = a(35) = 1 the cycle-length.
%t A099646 ed[x_] :=IntegerDigits[x]; f[x_] :=Apply[Plus, ed[x]^2]+1; itef[x_, ho_] :=NestList[f, x, ho]; tmc=Table[Length[Union[itef[w, 100]]], {w, 1, 256}]; c1=Table[Min[Flatten[Position[itef[w, Length[Union[itef[w, 100]]]] -Last[itef[w, Length[Union[itef[w, 100]]]]], 0]]], {w, 1, 256}]; (* transient-length= *) c1-1; (* cycle-length= *) c=tmc-(c1-1); (* ho=iteration number is chosen by trial and error *) (* program provides t, t+c and c lengths[=unknown-in-advance] for any similar iterations if f modified *)
%t A099646 (* Second program: *)
%t A099646 With[{nn = 10^3}, Table[Function[s, Length@ KeySelect[s, Length@ Lookup[s, #] > 1 &]]@ PositionIndex@ NestList[1 + Total[ IntegerDigits[#]^2] &, n, nn], {n, 105}]] (* _Michael De Vlieger_, Jul 24 2017 *)
%Y A099646 Cf. A007770, A031176, A099645, A099647.
%K A099646 nonn,base
%O A099646 1,1
%A A099646 _Labos Elemer_, Nov 11 2004