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.

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, 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, 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

Offset: 1

Author

Labos Elemer, Nov 11 2004

Keywords

• nonn
• base

Comments

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.

Examples

			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...
with t = 11 transient and c = a(1) = 9, the cycle-length;
For n = 35: list={36,46,53,[35],35,...} with transient t = 3, c = a(35) = 1 the cycle-length.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007770, A031176, A099645, A099647.

Programs

Mathematica

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 *)
(* Second program: *)
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 *)