A098189 -id:A098189 - cp's OEIS frontend

A098191 The values within a cycle of length 53 of the map x->A098189(x), sorted.

1500, 1570, 1714, 1718, 1722, 1768, 2062, 2066, 2070, 2084, 2120, 2220, 2276, 2328, 2348, 2578, 2582, 2586, 2760, 3218, 3222, 3328, 3428, 3552, 3704, 3736, 3792, 3864, 3984, 4192, 4324, 4332, 4400, 4480, 4656, 5088, 5128, 5464, 5544, 5856, 5872, 6200

Offset: 1

Labos Elemer, Sep 03 2004

See various attractors in A098191-A098195.

			The map has a trajectory 1500->2120->2084->1570->...->6660->9672->13248->11376->..->1768->1500 (returning
to the first term). The sequence contains all 53 members of this individual cycle, resorted to increasing magnitude.

Michael De Vlieger, Table of n, a(n) for n = 1..53

Cf. A034449, A000010, A063919, A098189-A098195.

Union@ NestList[Function[n, DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@ n], 1500, 53] (* Michael De Vlieger, Mar 01 2017 *)

Edited by R. J. Mathar, Mar 02 2009

A098195 Starting values x such that the map x -> A098189(x) enters any cycle of length 29.

Original entry on oeis.org

246, 250, 274, 276, 278, 282, 345, 356, 382, 386, 390, 392, 399, 400, 405, 424, 438, 468, 474, 478, 482, 484, 486, 490, 510, 522, 524, 534, 556, 562, 566, 570, 578, 579, 591, 594, 598, 602, 614, 618, 620, 621, 622, 626, 628, 630, 642, 645, 648, 650, 662

Offset: 1

Labos Elemer, Sep 03 2004

nonn

Iterating the map x -> A098189(x) may enter a cycle with 29 members (and there may be distinct cycles each with 29 members). The sequence lists all starting values of x such that (after some transient x) one of these cycles of length 29 is entered.
See other attractors and basins of attracted terms in A098191-A098195.
Corresponding number of transient terms for each term in a(n): {26, 25, 25, 25, 24, 23, 25, 26, 23, 22, 21, 24, 26, 24, 26, 25, 22, 27, 39, 17, 16, 22, 15, 27, 22, 27, 25, 25, 26, 16, 15, 14, 23, 25, 25, 33, 22, 39, 14, 13, 34, 26, 16, 15, 18, 14, 23, 34, 28, 20, 23, ...}. - Michael De Vlieger, Mar 01 2017

			282 is in the sequence since iterating the map x -> A098189(x) on that number yields 23 transient terms {282, 484, 390, 912, 1072, 628, 478, 482, 486, 570, 1296, 962, 1164, 1576, 998, 1002, 1684, 1270, 1800, 1860, 3360, 5568, 6008} then enters a cycle of 29 terms {3768, 4440, 7056, 6484, 4870, 6840, 9072, 8560, 7624, 4778, 4782, 7984, 4516, 3394, 3398, 3402, 4884, 7680, 10264, 6428, 4828, 4240, 3844, 2950, 3520, 3400, 2932, 2206, 2210}. - _Michael De Vlieger_, Mar 01 2017

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A063919, A098189, A098190, A098191, A098192, A098193, A098194.

Lookup[#, 29] &@ PositionIndex@ #[[All, -1]] &@ 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, 800}] (* Michael De Vlieger, Mar 01 2017, Version 10 *)

{x: A098190(x) = 29}.

Edited by R. J. Mathar, May 15 2009

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

Original entry on oeis.org

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

Labos Elemer, Sep 03 2004

nonn

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)

			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)

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

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

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 *)

Edited by R. J. Mathar, Mar 02 2009

A098192 Terms in a specific cycle of length 29 of the map x->A098189(x).

Original entry on oeis.org

2206, 2210, 2932, 2950, 3394, 3398, 3400, 3402, 3520, 3768, 3844, 4240, 4440, 4516, 4778, 4782, 4828, 4870, 4884, 6428, 6484, 6840, 7056, 7624, 7680, 7984, 8560, 9072, 10264

Offset: 1

Labos Elemer, Sep 03 2004

Iteration of the map x -> A098189(x) enters cycles of various lengths.
The 29 terms in the first cycle where A098190(x)=29 are listed here in ascending order; the example section of A098190 shows them in the mapping order.
See other attractors in A098191-A098195.

			The cycle follows the mapping 3768 -> 4440 -> ... -> 2210-> 3768 (returning to the first element).
This cycle is entered, for example, if the mapping is started at x=246 (Cf. A098190).

Cf. A063919, A098189, A098190, A098191, A098193, A098194, A098195.

Function[s, Union@ Drop[s, #] &@ Count[DeleteDuplicates@ Map[Position[s, #] &, s], k_ /; Length@ k == 1]]@ NestList[Function[n, DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@n], 246, 10^3] (* Michael De Vlieger, Mar 01 2017 *)

A098189(n)=my(f=factor(n)); prod(k=1,#f~, f[k,1]^f[k,2]+1) - eulerphi(f)
a(n)=if(n>1, A098189(n-1), 2206) \\ Charles R Greathouse IV, Mar 01 2017

Edited by R. J. Mathar, May 15 2009

A098193 Terms in a specific cycle of length 7 of the map x->A098189(x).

Original entry on oeis.org

1608, 1748, 1920, 2028, 2088, 2584, 2776

Offset: 1

Labos Elemer, Sep 03 2004

The map enters a cycle of length 7 if started at x=294, indicated by A098190(294)=7.
The group members of this cycle are listed here in ascending order.
See other attractors in A098191-A098195.

			An iteration started at 294 leads to a attractor which cycles through 7 numbers:
294->516->712->458->462->1032->1248->1464->1752->[2088->2028->2776->1748->1608->1920->2584->2088,..].
After 9 transients, the cycle (indicated by bracketing its members) is entered.

Cf. A063919, A098189, A098190, A098191, A098192, A098194, A098195.

A098189(n)=my(f=factor(n)); prod(k=1,#f~, f[k,1]^f[k,2]+1) - eulerphi(f)
a(n)=if(n>1, A098189(n-1), 1608) \\ Charles R Greathouse IV, Mar 01 2017

Edited by R. J. Mathar, May 15 2009

A098194 Starting values x such that the map x -> A098189(x) enters any cycle of length 53.

Original entry on oeis.org

46, 48, 50, 52, 54, 56, 58, 62, 66, 68, 69, 72, 82, 86, 88, 90, 93, 95, 96, 100, 106, 110, 112, 115, 117, 119, 123, 124, 132, 133, 135, 140, 141, 143, 145, 154, 155, 156, 158, 159, 162, 166, 168, 170, 174, 175, 176, 177, 180, 186, 187, 192, 195, 196, 200, 201

Offset: 1

Labos Elemer, Sep 03 2004

nonn

Iterating the map x -> A098189(x) may enter a cycle with 53 members
(and there may be distinct cycles each with 53 members). The sequence
lists all starting values of x such that (after some transient x) one of these cycles of length 53 is entered.

			None of the values of A098192, A098193 or A098195 is in the sequence, because these will enter or are already in cycles of length 29 or 7.

Cf. A063919, A098189, A098190, A098191, A098192, A098193, A098195.

{x: A098190(x) = 53}.

Edited by R. J. Mathar, May 15 2009

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A098191 The values within a cycle of length 53 of the map x->A098189(x), sorted.

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A098195 Starting values x such that the map x -> A098189(x) enters any cycle of length 29.

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

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A098192 Terms in a specific cycle of length 29 of the map x->A098189(x).

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A098193 Terms in a specific cycle of length 7 of the map x->A098189(x).

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A098194 Starting values x such that the map x -> A098189(x) enters any cycle of length 53.

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