A063919 -id:A063919 - cp's OEIS frontend

A097024 Numbers in the 5-cycle-attractor of the function f(x)=A063919(x).

1482, 1878, 1890, 2142, 2178

Offset: 1

Labos Elemer, Aug 30 2004

A002827 provides 1-cycle terms = unitary perfect numbers.
A063991 gives 2-cycle elements = unitary amicable numbers.
A097030 collects true 14-cycle elements, i.e., terms in end-cycle of length 14 when A063919(x) function is iterated.
Concerning 3-cycle elements, only {30,42,54} were encountered.

J. O. M. Pedersen, Order 5 cycles, 2007.

Cf. A063919, A002827, A063991, A097030.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a097024Q[k_] := Module[{a=NestList[a063919, k, 5]}, Count[a, k]==2&&Last[a]==k]
a097024[n_] := Select[Range[n], a097024Q]
a097024[2178] (* Hartmut F. W. Hoft, Jan 24 2024 *)

f(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - n;
isok5(n) = iferr(f(f(f(f(f(n))))) == n, E, 0);
isok1(n) = iferr(f(n) == n, E, 0);
isok(n) = !isok1(n) && isok5(n); \\ Michel Marcus, Sep 28 2018

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 *)

A063948 Unitary untouchable numbers: us(x) = n has no solution where us(x) (A063919) is the sum of the proper unitary divisors of x.

Original entry on oeis.org

2, 3, 4, 5, 7, 374, 702, 758, 998, 1542, 1598, 1778, 1808, 1830, 1974, 2378, 2430, 2910, 3164, 3182, 3188, 3216, 3506, 3540, 3666, 3698, 3818, 3846, 3986, 4196, 4230, 4574, 4718, 4782, 5126, 5324, 5610, 5738, 5918, 5952, 6002, 6174, 6270, 6404, 6450, 6510

Offset: 1

Felice Russo, Sep 04 2001

Pomerance & Yang show that this sequence has positive lower density (in fact, greater than 10^-7) and upper density at most 0.40632. - Charles R Greathouse IV, Dec 28 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
C. Pomerance and H.-S. Yang, On untouchable numbers and related problems, 2012
C. Pomerance and H.-S. Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation, to appear c. 2014

Cf. A034448, A063919, A064000.

us[x_] := us[x] = Total[ Select[ Divisors[x], GCD[#, x/#] == 1 &]] - x; us[1] = 1; usQ[n_] := With[{xm = Ceiling[n^2/4]}, Catch[ Do[ If[us[x] == n, Throw[True]]; If[x == xm, Throw[False]], {x, 1, xm}]]]; A063948 = Reap[ Do[ If[ !usQ[n], Print[n]; Sow[n]], {n, 1, 6600}]][[2, 1]] (* Jean-François Alcover, Jun 22 2012 *)

If us(x) = n > 1, then n^2 - 4x >= 0. - Dean Hickerson, Sep 04 2001.

More terms from David W. Wilson, Sep 05 2001

A318882 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n. Number of distinct terms in iteration list.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097032

nonn

This sequence implements the original definition given for A097032.

			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) = 0+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+1 = 2.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(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.
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..87360

Cf. A063919, A097031, A097032, A318883.

Cf. A002827 (the positions of ones after the initial 1).

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
a318882[n_] := Map[Length[NestWhileList[a063919, #, UnsameQ, All]]-1&, Range[n]]
a318882[105] (* Hartmut F. W. Hoft, Jan 25 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A318882(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(j-1), 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); };
A318882(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(j-1)); listput(visited, n); n = A063919(n)); };

a(n) = A097031(n) + A318883(n).

a(n) = A097032(n) + A318880(n) - 1.

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 *)

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]

A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097033

nonn

This sequence implements the original definition given for A097033.

			For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle (of length 1 in this case), thus there are no transient part, and a(1) = 0.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle 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) = 0, as 30, 42 and 54 are all contained in their own terminal cycle, without a preceding transient part.
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..87360

Cf. A003062, A063919, A097031, A097033, A318880, A318882.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
transient[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]]-1
a318883[n_] := Map[transient, Range[n]]
a318883[105] (* Hartmut F. W. Hoft, Jan 25 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n));
A318883(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-1), 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); };
A318883(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(k-1)); listput(visited, n); n = A063919(n)); };

a(n) = A318882(n) - A097031(n).

a(n) = A097033(n) + A318880(n) - 1.

A097034 Initial values for iteration of the function f(x) = A063919(x) such that the iteration ends in a 14-cycle, i.e., in A097030.

Original entry on oeis.org

1506, 1518, 1806, 1902, 1914, 1938, 1950, 2226, 2382, 2394, 2406, 2418, 2478, 2826, 2910, 2946, 2958, 3234, 3282, 3294, 3330, 3510, 3522, 3534, 3546, 3582, 3642, 3654, 3774, 3906, 3954, 3966, 3978, 4146, 4158, 4194, 4434, 4446, 4854, 4866, 4878, 5262

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			n=1506 is here because its iteration list = {1506, 1518, 1938, 2382, 2394, 2406, [2418, ...., 3582, 2418}. After a transient of length 6, the iteration ends in a cycle of length 14.

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
a097034Q[k_] :=
 Module[{iter = NestWhileList[a063919, k, UnsameQ, All]},
  Apply[Subtract, Reverse[Flatten[Position[iter, Last[iter]], 1]]] ==
   14]
a097034[n_] := Select[Range[n], a097034Q]
a097034[5262] (* Hartmut F. W. Hoft, Jan 25 2024 *)

cp's OEIS Frontend

A097024 Numbers in the 5-cycle-attractor of the function f(x)=A063919(x).

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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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A063948 Unitary untouchable numbers: us(x) = n has no solution where us(x) (A063919) is the sum of the proper unitary divisors of x.

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A318882 Total length of transient and terminal cycle if unitary-proper-divisor-sum function f(x) = A063919(x) is iterated and the initial value is n. Number of distinct terms in iteration list.

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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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