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

1,1

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.

Links

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

Example

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 = 1 cycle-length.

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 *)

Crossrefs

Cf. A007770, A031176, A099645, A099647.

Sequence in context: A292864 A363120 A363021 * A181693 A271880 A180599

Adjacent sequences: A099643 A099644 A099645 * A099647 A099648 A099649

Keyword

nonn,base

Author

Labos Elemer, Nov 11 2004

Status

approved