A063919 -id:A063919 - 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

A097033 Number of transient terms before either 0 or a finite cycle is reached when unitary-proper-divisor-sum-function f(x) = A034460(x) is iterated and the initial value is n.

Original entry on oeis.org

1, 2, 2, 2, 2, 0, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 2, 4, 2, 4, 3, 5, 2, 4, 2, 3, 2, 4, 2, 0, 2, 2, 4, 5, 3, 5, 2, 6, 3, 5, 2, 0, 2, 3, 4, 4, 2, 5, 2, 5, 4, 5, 2, 0, 3, 3, 3, 3, 2, 0, 2, 6, 3, 2, 3, 2, 2, 6, 3, 7, 2, 5, 2, 6, 3, 5, 3, 1, 2, 6, 2, 4, 2, 6, 3, 5, 5, 5, 2, 0, 4, 5, 4, 6, 3, 6, 2, 6, 4, 1, 2, 1, 2, 6, 6

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			From _Antti Karttunen_, Sep 24 2018: (Start)
For n = 1, A034460(1) = 0 that is, we end to a terminal zero after a transient part of length 1, thus a(1) = 1.
For n = 2, A034460(2) = 1, and A034460(1) = 0, so we end to a terminal zero after a transient part of length 2, thus a(2) = 2.
For n = 30, A034460(30) = 42, A034460(42) = 54, A034460(54) = 30, thus a(30) = a(42) = a(54) = 0, as 30, 42 and 54 are all contained in their own terminal cycle, without a preceding transient part. (End)
For n = 1506, the iteration-list is {1506, 1518, 1938, 2382, 2394, 2406, [2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 2418, ..., ad infinitum]}. After a transient of length 6 the iteration ends in a cycle of length 14, thus a(1506) = 6.
If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A002827, A003062, A034460, A063919, A063991, A097024, A097030, A097031, A097032-A097037, A318880.

Cf. A318883 (sequence that implements the original definition of this sequence).

a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>0
transient[k_] := Module[{iter=NestWhileList[a034460, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]-1]
a097033[n_] := Map[transient, Range[n]]
a097033[105] (* Hartmut F. W. Hoft, Jan 24 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A097033(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-1), mapput(visited, n, j)); n = A034460(n); if(!n,return(j))); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A318883(n) + (1-A318880(n)). - Antti Karttunen, Sep 23 2018

Definition corrected (to agree with the given terms) by Antti Karttunen, Sep 23 2018, based on observations by Hartmut F. W. Hoft

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

A319917 Unitary sociable numbers of order six.

Original entry on oeis.org

698130, 698310, 698490, 712710, 712890, 713070, 341354790, 348612390, 391662810, 406468314, 411838938, 519891750, 530946330, 582129630, 596171970, 621549630, 717175170, 740700270, 740700450, 743324934, 838902150, 919121658, 1009954170, 1343332998

Offset: 1

Michel Marcus, Oct 01 2018

nonn

Note that the first 6 terms and the next 6 terms form two sociable groups. But then the next 12 terms belong to two distinct sociable groups.

J. O. M. Pedersen, Known Unitary Sociable Numbers of order different from four [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Order 6 cycles, 2007.
Eric Weisstein's World of Mathematics, Unitary Sociable Numbers

Cf. A063919 (sum of proper unitary divisors).
Cf. A002827 (unitary perfect), A063991 (unitary amicable).
Cf. A319902 (order 4), A097024 (order 5), A097030 (order 14).

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok6(n) = iferr(f(f(f(f(f(f(n)))))) == n, E, 0);
isok3(n) = iferr(f(f(f(n))) == n, E, 0);
isok2(n) = iferr(f(f(n)) == n, E, 0);
isok1(n) = iferr(f(n) == n, E, 0);
isok(n) = isok6(n) && !isok1(n) && !isok2(n) && !isok3(n);

A063919(n) = my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + 1) - n
is(n) = my(c = n); for(i = 1, 5, c = A063919(c); if(c == 1 || c == n, return(0))); c = A063919(c); c == n \\ David A. Corneth, Oct 01 2018

A319937 Unitary sociable numbers of order 10.

Original entry on oeis.org

525150234, 527787366, 528544218, 553128198, 612951066, 675192294, 735821562, 982674438, 998151162, 998151174, 5251502340, 5277873660, 5285442180, 5531281980, 6129510660, 6751922940, 7358215620, 9826744380, 9981511620, 9981511740

Offset: 1

Michel Marcus, Oct 02 2018

J. O. M. Pedersen, Known Unitary Sociable Numbers of order different from four [Via Internet Archive Wayback-Machine]
Eric Weisstein's World of Mathematics, Unitary Sociable Numbers
J. O. M. Pedersen, Order 10 cycles

Cf. A063919 (sum of proper unitary divisors).
Cf. A002827 (unitary perfect), A063991 (unitary amicable).
Cf. A097024 (order 5), A319917 (order 6), A097030 (order 14).

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok10(n) = iferr(f(f(f(f(f(f(f(f(f(f(n)))))))))) == n, E, 0);
isok5(n) = iferr(f(f(f(f(f(n))))) == n, E, 0);
isok2(n) = iferr(f(f(n)) == n, E, 0);
isok1(n) = iferr(f(n) == n, E, 0);
isok(n) = isok10(n) && !isok1(n) && !isok2(n) && !isok5(n);

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

Original entry on oeis.org

6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321

Offset: 1

Amiram Eldar, May 29 2020

nonn

Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
The squarefree terms of A335267 are also terms of this sequence.
The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.

			6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..1000

The unitary version of A335267.
A002827 is subsequence.
Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]

cp's OEIS Frontend

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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A097033 Number of transient terms before either 0 or a finite cycle is reached when unitary-proper-divisor-sum-function f(x) = A034460(x) is iterated and the initial value is n.

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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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A319917 Unitary sociable numbers of order six.

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