A002827 -id:A002827 - cp's OEIS frontend

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.

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]

A098186 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 0 or belongs to A002827, a unitary-perfect-number >1: 6,60,90,87360... Sequence gives initial values for which the iteration ends in 87360, the 4th unitary perfect number.

Original entry on oeis.org

87360, 232608, 356640, 465144, 527712, 565728, 713208, 1018248, 1055352, 1211352, 1240032, 1303728, 1316904, 1352568, 1357584, 1360416, 1379280, 1550472, 1690440, 1835592, 2035608, 2078328, 2110632, 2262892, 2422632

Offset: 1

Labos Elemer, Aug 31 2004

nonn

			Iteration list started from n=1018248: {1018248, 1055352, 527712, 232608, 87360, 87360...}

Cf. A063919, A002827, A063991, A097024, A097030-A097037, A098185.

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, 87360], Print[{n, s}]; ta=Append[ta, n]], {n, 1, 5000000}];ta = Delete[ta, 1]

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

Original entry on oeis.org

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

A129485 Odd unitary abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035

Offset: 1

Ant King, Apr 17 2007

This sequence is different from A112643. The two sequences agree for the first 50 terms but differ thereafter. The exceptions, i.e. those odd unitary abundant numbers that are not squarefree ordinary abundant numbers, are in A129486.
22309287 is the smallest term not divisible by 5. 33426748355 is the smallest term not divisible by 3. - Donovan Johnson, May 15 2013
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 137, 1714, 16918, 181744, 1752337, 17290556, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00017... . - Amiram Eldar, Sep 02 2022

			The third odd unitary abundant number is 21945. Hence a(3) = 21945.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034683, A129486, A034460, A034448, A129487, A002827, A129468, A112643.

# see A034683 for the code of isA034683()
isA129485 := proc(n)
    type(n,'odd') and isA034683(n) ;
end proc:
for n from 1 do
    if isA129485(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;Select[Range[1,10^5,2],sstar[ # ]># &]

This sequence contains the odd members of A034683. i.e. odd numbers with a positive unitary abundance (A129468).

A290466 Unitary Zumkeller numbers: numbers k whose unitary divisors can be partitioned into two disjoint subsets whose sums are both usigma(k)/2.

Original entry on oeis.org

6, 30, 42, 60, 66, 70, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834, 840, 858, 870, 894, 906

Offset: 1

Ivan N. Ianakiev, Aug 03 2017

nonn

Seemingly, a supersequence of A002827 (unitary perfect numbers) and a subsequence of A083207 (Zumkeller numbers).

			The set of unitary divisors of 30 is {1,2,3,5,6,10,15,30}. It can be partitioned into two disjoint subsets with equal sums of elements: {5,6,10,15} and {1,2,3,30}, therefore 30 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bhabesh Das, On unitary Zumkeller numbers, Notes on Number Theory and Discrete Mathematics, Vol. 30, No. 2 (2024), pp. 436-442.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.

Cf. A002827, A034448 (sum of unitary divisors of n), A083207, A290467.

uDiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]];uZNQ[n_]:=Module[{d=uDiv[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]];Select[Range[10^3],uZNQ] (* combined from the code by Robert G. Wilson v at A034448 and T. D. Noe at A083207 *)

A293188 Unitary pseudoperfect numbers: numbers that equal to the sum of a subset of their aliquot unitary divisors.

Original entry on oeis.org

6, 30, 42, 60, 66, 78, 90, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Supersequence of A002827.

The nonsquarefree terms are 60, 90, 150, 294, 420, 630, 660, 726, 750, 780, 840, ...

			150 is in the sequence since its unitary aliquot divisors are 1, 2, 3, 6, 25, 50, 75 and 150 = 25 + 50 + 75.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002827, A005835, A034683, A064114.

udiv[n_]:=Block[{d=Divisors[n]},Select[d,GCD[#,n/#]==1&]]; a={};n=0;While[Length[a]<100,n++;d=Most[udiv[n]];c = SeriesCoefficient[ Series[ Product[1+x^d[[i]],{i,Length[d]} ],{x,0,n}], n]; If[c>0,AppendTo[a,n]]];a (* after T. D. Noe at A005835 *)

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

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.

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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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A129485 Odd unitary abundant numbers.

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A290466 Unitary Zumkeller numbers: numbers k whose unitary divisors can be partitioned into two disjoint subsets whose sums are both usigma(k)/2.

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A293188 Unitary pseudoperfect numbers: numbers that equal to the sum of a subset of their aliquot unitary divisors.

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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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