A098190 The length of the cycle reached for the map x->A098189(x) if started at n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 5, 53, 1, 53, 1, 53, 5, 53, 1, 53, 1, 53, 5, 53, 1, 5, 1, 53, 5, 1, 5, 53, 1, 53, 53, 5, 1, 53, 1, 5, 1, 1, 5, 5, 1, 5, 1, 53, 1, 5, 5, 53, 1, 53, 1, 53, 5, 1, 53, 5, 53, 53

Offset: 1

Author

Labos Elemer, Sep 03 2004

Keywords

• nonn

Comments

See various attractors in A098191-A098195.

For n below 10^6, cycle-lengths are one of {1,2,3,4,5,6,7,8,9,14,18,20,29,32,47,53}.

From Michael De Vlieger, Mar 02 2017: (Start)

Corresponding number of transient terms: {0, 0, 1, 2, 1, 5, 1, 2, 3, 4, 1, 5, 1, 3, 5, 4, 1, 2, 1, 6, 7, 5, 1, 1, 6, 4, 5, 0, 1, 3, 1, 2, 1, 19, 2, 19, 1, 18, 3, 19, 1, 17, 1, 20, 20, 49, 1, 51, 3, 48, 20, 50, 1, 46, 3, 52, 21, 47, 1, 13, 1, 46, 21, 2, 20, 45, 1, 48, 51, 24, 1, 46, 1, 12, 3, 3, 20, 11, 1, 25, 1, 44, 1, 16, 21, 43, 3, 49, 1, 42, 20, 4, 49, 15, 52, 44, ...}.

Maximum number of transient terms for n = 2^m: {0, 0, 2, 5, 5, 7, 52, 53, 53, 53, 53, 68, 73, 89, 164, 197, 213, 241, 372, 422, ...}.

Maximum number of transient terms for n = 10^m: {0, 5, 52, 53, 89, 235, 502, ...}.

(End)

Examples

			Starting at n=10, the trajectory is 10->14->18->24->28->28->28 (repeating), so the cycle has length a(10)=1.
Starting at n=246, the trajectory is 246->424->278..->6008->[3768->4440->...,10264,6428,...->2206->2210->3768], where the cycle of length a(246)=29 has been put into brackets.
From _Michael De Vlieger_, Mar 01 2017: (Start)
a(746)=3 since the trajectory is 746->750->1312->746 (repeating).
a(3238)=4 since the trajectory begins with transient terms {3238, 3242, 3246, 5424, 5960, 5732, 4306, 4310, 6056, 3798, 5100}, followed by the cycle {8080, 7204, 5410, 7596}.
Statistics regarding a(n) for 1<=n<=10^6:
Cycle    | Least n with | Frequency of cycle length for n <=
length   | cycle length | 10^4    10^5     10^6
   1            1         1337    9756    78784
   2         1186           39     147      521
   3          746            6      14       17
   4         3238           43     127      430
   5           34          722    1375     1740
   6         2226          231    3285    19368
   7          294          707    3782    39384
   8         5306           44    1892    21583
   9         1806          175     696     2269
  14         9902            2    2256    53777
  18        14422            0    2013    46218
  20         9026            3    5271    67258
  29          246         3709   35454   239197
  32        11802            0    1342     8321
  47        19554            0    1838   109448
  53           46         2982   30752   311685
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A034448, A000010, A063919, A098189-A098195.

Programs

Mathematica

Last /@ Table[If[n == 1, {0, 1}, Function[s, Function[t, {#, First@ Differences@ Take[Flatten@ t[[# + 1]], 2]} &@ Count[DeleteDuplicates@ t, k_ /; Length@ k == 1]]@ Map[Position[s, #] &, s]]@ NestList[Function[n, DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@ n],n, n + 120]], {n, 96}] (* or, faster *)
f[n_] := Module[{s = {n}, k, g}, g[x_] := DivisorSum[x, # &, CoprimeQ[#, x/#] &] - EulerPhi@ x; k = g@ n; While[Count[s, k] <= 1, AppendTo[s, k]; k = g@ Last@ s]; s]; Table[If[n == 1, {0, 1}, Function[s, Function[t, {#, First@ Differences@ Take[Flatten@ t[[# + 1]], 2]} &@ Count[DeleteDuplicates@ t, k_ /; Length@ k == 1]]@ Map[Position[s, #] &, s]]@ f@ n], {n, 96}] (* Michael De Vlieger, Mar 01 2017 *)

Extensions

Edited by R. J. Mathar, Mar 02 2009