A097024 -id:A097024 - cp's OEIS frontend

A097035 Initial values for the iteration of the function f(x) = A063919(x) such that the iteration ends in a 5-cycle, i.e., in A097024.

570, 870, 1230, 1290, 1326, 1482, 1530, 1686, 1698, 1710, 1794, 1866, 1878, 1890, 2058, 2070, 2142, 2154, 2166, 2178, 2238, 2250, 2502, 2802, 2814, 3042, 3222, 3630, 3702, 3714, 3726, 4350, 4494, 4506, 4518, 4914, 5010, 5142, 5154, 5166, 5284, 5418

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			n = 570: list = {570, 870, 1290, [1878, 1890, 2142, 2178, 1482], 1878}; after 3 transients, a 5-cycle arises.
n = 1230: {1230, 1794, 2238, 2250, 1530, 1710, [1890, 2142, 2178, 1482, 1878]} ; the iteration to the 5-cycle is not necessarily monotone. - _Hartmut F. W. Hoft_, Jan 25 2024

Cf. A002827, A063919, A063991, A097024, A097030-A097037.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a097035Q[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Apply[Subtract, Reverse[Flatten[Position[iter, Last[iter]], 1]]]==5]
a097035[n_] := Select[Range[n], a097035Q]
a097035[5418] (* Hartmut F. W. Hoft, Jan 25 2024 *)

A097030 Numbers in the cycle-attractors of length=14 of the function f(x)=A063919(x).

Original entry on oeis.org

2418, 2958, 3522, 3534, 3582, 3774, 3906, 3954, 3966, 3978, 4146, 4158, 4434, 4446, 24180, 29580, 35220, 35238, 35340, 35820, 37740, 38682, 39060, 39540, 39660, 39780, 41460, 41580, 44340, 44460, 45402, 49878, 65190, 65322, 74430, 74610, 74790, 98106, 101478, 117258, 117270, 117450

Offset: 1

Labos Elemer, Aug 30 2004

nonn

This sequence collects 14-cycle-attractor elements for iteration of sum-proper-unitary-divisors.
A002827 provides 1-cycle terms = unitary perfect numbers.
A063991 gives 2-cycle elements = unitary amicable numbers.
A097024 collects true 5-cycle elements, i.e., terms in end-cycle of length 5 when A063919(x) function is iterated.
Concerning 3-cycle elements, only {30,42,54} were encountered.

			These 42 numbers are in 3 different 14-cycles. The first is: [2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582]. [edited by _Michel Marcus_, Sep 29 2018]

J. O. M. Pedersen, Known Unitary Sociable Numbers of order different from four [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Order 14 cycles, 2007.

Cf. A063919, A002827, A063991, A097024.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a097030Q[k_] := Module[{a=NestList[a063919, k, 14]}, Count[a, k]==2&&Last[a]==k]
a097030[n_] := Select[Range[n], a097030Q]
a097030[117450] (* Hartmut F. W. Hoft, Jan 24 2024 *)

More terms from Michel Marcus, Sep 29 2018

A097037 Initial values for iteration of A063919(x) function such that iteration ends in a 2-cycle, i.e., "attracted" by unitary amicable numbers, A063991.

Original entry on oeis.org

102, 114, 126, 388, 436, 484, 812, 866, 1020, 1036, 1040, 1116, 1140, 1196, 1260, 1380, 1500, 1524, 1532, 1552, 1618, 1644, 1716, 1724, 1726, 1744, 1916, 2020, 2066, 2068, 2324, 2368, 2386, 2486, 2592, 2684, 2880, 2924, 3032, 3098, 3120, 3124, 3136

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			n = 866: list = {866, 436, 114, 126, 114} ending in 2-cycle of [114, 126].

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

Cf. A002827, A063919, A063991, A097024, A097030-A097036.

s[n_] := If[n > 1, Times @@ (1 + Power @@@ FactorInteger[n]) - n, 0]; useq[n_] := Most[NestWhileList[s, n, UnsameQ, All]]; amiQ[k_] := Module[{s1 = s[k]}, s1 != k && s[s1] == k]; aQ[n_] := amiQ[Last[useq[n]]]; Select[Range[5000], aQ] (* Amiram Eldar, Apr 06 2019 *)

A097032 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A034460(x) is iterated and the initial value is n. Number of distinct terms in iteration list, including also the terminal 0 in the count if the iteration doesn't end in a cycle.

Original entry on oeis.org

2, 3, 3, 3, 3, 1, 3, 3, 3, 4, 3, 4, 3, 5, 4, 3, 3, 5, 3, 5, 4, 6, 3, 5, 3, 4, 3, 5, 3, 3, 3, 3, 5, 6, 4, 6, 3, 7, 4, 6, 3, 3, 3, 4, 5, 5, 3, 6, 3, 6, 5, 6, 3, 3, 4, 4, 4, 4, 3, 1, 3, 7, 4, 3, 4, 3, 3, 7, 4, 8, 3, 6, 3, 7, 4, 6, 4, 2, 3, 7, 3, 5, 3, 7, 4, 6, 6, 6, 3, 1, 5, 6, 5, 7, 4, 7, 3, 7, 5, 4, 3, 3, 3, 7, 7

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			From _Antti Karttunen_, Sep 24 2018: (Start)
For n = 1, A034460(1) = 0, thus a(1) = 1+1 = 2.
For n = 2, A034460(2) = 1, and A034460(1) = 0, so we end to the zero after a transient part of length 2, thus a(2) = 2+1 = 3.
For n = 30, A034460(30) = 42, A034460(42) = 54, A034460(54) = 30, thus a(30) = a(42) = a(54) = 0+3 = 3, as 30, 42 and 54 are all contained in their own terminal cycle of length 3, 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+14 = 20.

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

Cf. A003062, A034460, A063919, A063991, A097024, A097030, A097031, A097033-A097037, A318880, A318882.
Cf. A002827 (the positions of ones).
Cf. A318882 (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
a097032[n_] := Map[Length[NestWhileList[a034460, #, UnsameQ, All]]-1&, Range[n]]
a097032[105] (* Hartmut F. W. Hoft, Jan 24 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A097032(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-1), mapput(visited, n, j)); n = A034460(n); if(!n,return(j+1))); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A318882(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

A097031 Length of terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is 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, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			From _Antti Karttunen_, Sep 22 2018: (Start)
For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle of length 1 without a preceding transient part, thus a(1) = 1.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle of length 1 (after a transient part of length 1) thus a(2) = 1.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 3, as 30, 42 and 54 are all contained in their own terminal cycle of length 3, 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) = 14.

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

Cf. A002827, A063919, A063991, A097024, A097030, A097032, A097033, A097034, A097035, A097036, A097037, A318882, A318883.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
cycleLength[k_] := Module[{cycle=NestWhileList[a063919, k, UnsameQ, All]}, Apply[Subtract, Reverse[Flatten[Position[cycle, Last[cycle]], 1]]]]
a097031[n_] := Map[cycleLength, Range[n]]
a097031[105] (* Hartmut F. W. Hoft, Jan 24 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A097031(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-mapget(visited, n)), mapput(visited, n, j)); n = A063919(n)); };
\\ Or by using lists:
pil(item,lista) = { for(i=1,#lista,if(lista[i]==item,return(i))); (0); };
A097031(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(j-k)); listput(visited, n); n = A063919(n)); }; \\ Antti Karttunen, Sep 22 2018

a(n) = A318882(n) - A318883(n). - Antti Karttunen, Sep 22 2018

A097036 Initial values for iteration of A063919[x] function such that iteration ends in a 3-cycle.

Original entry on oeis.org

30, 42, 54, 100, 140, 148, 194, 196, 208, 220, 238, 252, 274, 288, 300, 336, 348, 350, 364, 374, 380, 382, 386, 400, 420, 440, 492, 516, 528, 540, 542, 550, 592, 600, 612, 660, 694, 708, 720, 722, 740, 756, 758, 764, 766, 780, 792, 794, 820, 836, 900, 932

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			n=100: list={100, [30, 42, 54], 30, ... after 1 transient a 3-cycle arises.

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

Cf. A002827, A063919, A063991, A097024, A097030-A097037.

s[n_] := If[n > 1, Times @@ (1 + Power @@@ FactorInteger[n]) - n, 0]; useq[n_] := Most[NestWhileList[s, n, UnsameQ, All]]; cycleQ[k_] := Module[{s1 = s[k]}, s1 != k && s[s[s1]] == k]; aQ[n_] := cycleQ[Last[useq[n]]]; Select[Range[1000], aQ] (* Amiram Eldar, Apr 06 2019 *)

A003062 Beginnings of periodic unitary aliquot sequences.

Original entry on oeis.org

6, 30, 42, 54, 60, 66, 78, 90, 100, 102, 114, 126, 140, 148, 194, 196, 208, 220, 238, 244, 252, 274, 288, 292, 300, 336, 348, 350, 364, 374, 380, 382, 386, 388, 400, 420, 436, 440, 476, 482, 484, 492, 516, 528, 540, 542, 550, 570, 578, 592, 600, 612, 648, 660, 680, 688, 694, 708, 720, 722, 740, 756, 758, 764, 766, 770, 780, 784, 792, 794, 812

Offset: 1

N. J. A. Sloane

nonn

Provided that A034460 has no infinite unbounded trajectories, these are also numbers m such that when iterating the map k -> A034460(k), starting from k = m, the iteration will never reach 0, that is, will instead eventually enter into a finite cycle. - Antti Karttunen, Sep 23 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000
H. J. J. Te Riele, Unitary Aliquot Sequences, Report MR-139/72, Mathematisch Centrum, Amsterdam, September 1972.

Cf. A034460, A097010 (complement), A318880 (characteristic function).

Cf. also A002827, A063991, A097024, A097030, A097034, A097035, A097036, A097037.

a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>0
periodicQ[k_] := NestWhile[a034460, k, UnsameQ, All]!=0
nmax = 812; Select[Range[nmax], periodicQ]
(* Hartmut F. W. Hoft, Jan 24 2024 *)

up_to = 20000;
A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A318880(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(1), mapput(visited, n, j)); n = A034460(n); if(!n,return(0))); };
A003062list(up_to) = { my(v = vector(up_to), k=0, n=1); while(kA318880(n), k++; v[k] = n); n++); (v); };
v003062 = A003062list(up_to);
A003062(n) = v003062[n]; \\ Antti Karttunen, Sep 23 2018

More terms from Antti Karttunen, Sep 23 2018

A097023 Numbers k such that the sum of the divisors of 2*k^2 is a square.

Original entry on oeis.org

313, 335, 2612, 2817, 3015, 3820, 23508, 34380, 36647, 38193, 42217, 50281, 64972, 73535, 96404, 103180, 155991, 265364, 325847, 329823, 379953, 397068, 452529, 476545, 584748, 624748, 661815, 668660, 867636, 928620

Offset: 1

Labos Elemer, Aug 24 2004

nonn

			sigma(2*313^2) = 543^2.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000203, A008847, A074388, A097024.

Select[Range[950000],IntegerQ[Sqrt[DivisorSigma[1,2#^2]]]&] (* Harvey P. Dale, Jul 10 2012 *)

a(n) = sqrt(A074388(n)/2). - Amiram Eldar, Aug 13 2024

Definition clarified by Harvey P. Dale, Jul 10 2012

A098185 If f(x) = (sum of unitary proper divisors of x) = A063919(x) is iterated, the iteration may lead to a fixed point which is either equals 0 or it is from A002827, a unitary perfect number > 1: 6,60,90,87360... Here initial values are collected for which the iteration ends in a unitary perfect number > 1.

Original entry on oeis.org

6, 60, 66, 78, 90, 244, 292, 476, 482, 578, 648, 680, 688, 770, 784, 832, 864, 956, 958, 976, 1168, 1354, 1360, 1392, 1488, 1600, 1658, 1670, 1906, 2232, 2264, 2294, 2376, 2480, 2552, 2572, 2576, 2626, 2712, 2732, 2806, 2842, 2870, 2904, 2912, 2992, 3024

Offset: 1

Labos Elemer, Aug 31 2004

nonn

			Initial values attracted by 87360 (4th unitary perfect number) are collected separately in A098186.
It seems that 6 is the only initial value ending in fixed point = 6.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A063919, A002827, A063991, A097024, A097030-A097037.

di[x_] :=Divisors[x];ta={{0}}; ud[x_] :=Part[di[x],Flatten[Position[GCD[di[x],Reverse[di[x]]],1]]]; asu[x_] :=Apply[Plus,ud[x]]-x;nsf[x_,ho_] :=NestList[asu,x,ho] Do[g=n;s=Last[NestList[asu,n,100]]; If[Equal[s,6]||Equal[s,60]||Equal[s,90],Print[{n,s}]; ta=Append[ta,n]],{n,1,256}];ta = Delete[ta,1]

A319902 Unitary sociable numbers of order 4.

Original entry on oeis.org

263820, 263940, 280380, 280500, 395730, 395910, 420570, 420750, 172459210, 209524210, 218628662, 218725430, 230143790, 231439570, 246667790, 272130250, 384121920, 384296640, 408233280, 408408000

Offset: 1

Michel Marcus, Oct 01 2018

Is this a duplicate of A098188? - R. J. Mathar, Oct 04 2018
Note that the first 4 terms and the next 4 terms form two sociable groups. But then the next 8 terms belong to two distinct sociable groups, whereas in A098188 the integers are grouped by cycle.
From Hartmut F. W. Hoft, Aug 23 2023: (Start)
This sequence is A098188 in ascending order.
Among the 19 4-cycles listed in the link by J. O. M. Pedersen only four of the 6 possible patterns of relative sizes of the numbers in a cycle are realized. (End)

J. O. M. Pedersen, Known Unitary Sociable Numbers of order four [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Order 4 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. A097024 (order 5), A097030 (order 14).
Cf. A090615 (least member of sociable quadruples).
Cf. A098188.

f[n_] := f[n] = Module[{s = 0}, s = Total[Select[Divisors[n], GCD[#, n/#] == 1 &]]; Return[s - n]]; isok1[n_] := isok1[n] = Quiet[Check[f[n] == n, 0]]; isok2[n_] := isok2[n] = Quiet[Check[f[f[n]] == n, 0]]; isok4[n_] := isok4[n] = Quiet[Check[f[f[f[f[n]]]] == n, 0]]; isok[n_] := isok[n] = isok4[n] && Not[isok1[n]] && Not[isok2[n]]; Monitor[Position[Table[isok[n], {n, 1, 408408000}], True], n] (* Robert P. P. McKone, Aug 24 2023 *)

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok4(n) = iferr(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) = isok4(n) && !isok1(n) && !isok2(n);

cp's OEIS Frontend

A097035 Initial values for the iteration of the function f(x) = A063919(x) such that the iteration ends in a 5-cycle, i.e., in A097024.

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A097030 Numbers in the cycle-attractors of length=14 of the function f(x)=A063919(x).

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A097037 Initial values for iteration of A063919(x) function such that iteration ends in a 2-cycle, i.e., "attracted" by unitary amicable numbers, A063991.

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A097032 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A034460(x) is iterated and the initial value is n. Number of distinct terms in iteration list, including also the terminal 0 in the count if the iteration doesn't end in a cycle.

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A097031 Length of terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n.

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A097036 Initial values for iteration of A063919[x] function such that iteration ends in a 3-cycle.

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A003062 Beginnings of periodic unitary aliquot sequences.

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A097023 Numbers k such that the sum of the divisors of 2*k^2 is a square.

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