A063919 -id:A063919 - 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 *)

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]

A098188 Irregular triangle with 4 columns which contains in each row the members of a 4-cycle under the map x->A063919(x), iteration of summing the proper-unitary divisors.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Sep 02 2004

Initial values attracted by this sequence are in A098187.

The iteration of this function also contains 2-cycles like 114->126->114... or 1140 -> 1260 ->1140,... or 3-cycles like 30->42->54->30->....

			The first line represents the 4-cycle  280500->263820->263940->280380->280500->...,
The second line represents the 4-cycle 420750->395730->395910->420570->420750->..

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. A098185, A098186, A002827, A097024, A097030, A098187.

Cf. A319902 (where the terms are entered by increasing value).

More terms from Michel Marcus, Oct 05 2018

A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

Original entry on oeis.org

81570, 114270, 137046, 169998, 177906, 182082, 182094, 185190, 194574, 194586, 201642, 203442, 204420, 204540, 212466, 212874, 213870, 219306, 219318, 230874, 231438, 231834, 231846, 232626, 237678, 238134, 242634, 258882, 259338, 259350

Offset: 1

Labos Elemer, Sep 02 2004

nonn

The sequence is the attractor-basin of set of {C4} cycles belonging to this iteration.

The {C4} attractor-set is displayed separately in A098188.

			81570 is in the sequence because its track under the iterated map is 81570, 114270, 182082, 182094, 232626, 237678, 305682, 352878, 360978, 403662, 420738, [420750, 395730, 395910, 420570], 420750.., where the cycle is indicated by brackets. The 4 recurrent terms appear after 11 transients for this case.

Cf. A098185, A098186, A002827, A097024, A097030.

A332974 Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = usigma(k) - k is the sum of proper unitary divisors of k (A063919).

Original entry on oeis.org

3, 21, 321, 1257, 3237, 146139, 268713, 584835, 26749089, 9988999095, 25997557299, 54449485353, 935628578283, 2105722150095, 3921293253003, 8234992646643

Offset: 1

Amiram Eldar, Mar 04 2020

a(17) > 10^13. - Giovanni Resta, May 09 2020

			21 is a term since s(21) = 11 and s(19) + s(20) = 1 + 10 = 11.

The unitary version of A291176.

Cf. A063919, A065557, A065900, A075565, A076136, A145469, A291126, A292033, A294995, A332973.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s[n_] := usigma[n] - n; Select[Range[3, 6*10^5], s[#] == s[# - 1] + s[# - 2] &]

a(12)-a(16) from Giovanni Resta, May 09 2020

A369895 Irregular triangle of iteration steps of A063919 until the end of the terminal cycle is reached, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 7, 1, 8, 1, 9, 1, 10, 8, 1, 11, 1, 12, 8, 1, 13, 1, 14, 10, 8, 1, 15, 9, 1, 16, 1, 17, 1, 18, 12, 8, 1, 19, 1, 20, 10, 8, 1, 21, 11, 1, 22, 14, 10, 8, 1, 23, 1, 24, 12, 8, 1, 25, 1, 26, 16, 1, 27, 1, 28, 12, 8, 1, 29, 1, 30, 42, 54

Offset: 1

Hartmut F. W. Hoft, Feb 04 2024

			The beginning of the irregular triangle showing 3 terminal cycles ( 1 ), ( 6 ) and ( 30 42 54 ):
  1
  2    1
  3    1
  4    1
  5    1
  6
  7    1
  ...
  14  10   8   1
  ...
  30  42  54
  31  1
  ...
Row 1230 contains a non-monotone iteration that ends in the 5-cycle starting at A097035(3):
1230, 1794, 2238, 2250, 1530, 1710, {1890, 2142, 2178, 1482, 1878 }.

Cf. A002827, A063919, A097024, A097030, A097031, A097034, A097035, 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
iter[k_] := Most[NestWhileList[a063919, k, UnsameQ, All]]
a369895[n_] := Map[iter, Range[n]]
a369895[30] (* irregular triangle *)
Flatten[a369895[30]] (* sequence data *)

A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 12, 1, 16, 1, 12, 1, 42, 1, 1, 15, 20, 13, 14, 1, 22, 17, 14, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 30, 17, 16, 23, 32, 1, 60, 1, 34, 17, 1, 19, 78, 1, 22, 27, 74, 1, 18, 1, 40, 29, 24, 19, 90, 1, 22, 1, 44

Offset: 1

N. J. A. Sloane

			Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation 83.288 (2014): 1903-1913; alternative link.

Cf. A034444, A034448, A077610, A306633.
Cf. A063936 (squares > 1).
Cf. A063919 (essentially the same sequence).

a034460 = sum . init . a077610_row  -- Reinhard Zumkeller, Aug 15 2012

A034460 := proc(n)
    A034448(n)-n ;
end proc:
seq(A034460(n),n=1..40) ; # R. J. Mathar, Nov 10 2014

usigma[n_] := Sum[ If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := usigma[n] - n; Table[ a[n], {n, 1, 82}] (* Jean-François Alcover, May 15 2012 *)
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 03 2022 *)

a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ Charles R Greathouse IV, Aug 01 2016

a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - Reinhard Zumkeller, Aug 15 2012

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - Amiram Eldar, Feb 22 2024

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

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);

A098189 Sum of unitary divisors minus Euler phi: a(n) = A034448(n) - A000010(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 10, 2, 5, 4, 14, 2, 16, 2, 18, 16, 9, 2, 24, 2, 22, 20, 26, 2, 28, 6, 30, 10, 28, 2, 64, 2, 17, 28, 38, 24, 38, 2, 42, 32, 38, 2, 84, 2, 40, 36, 50, 2, 52, 8, 58, 40, 46, 2, 66, 32, 48, 44, 62, 2, 104, 2, 66, 44, 33, 36, 124, 2, 58, 52, 120, 2, 66, 2, 78, 64, 64, 36, 144, 2

Offset: 1

Labos Elemer, Sep 03 2004

nonn

			a(1) = 1 - 1 = 0.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034448, A000010, A063919, A098190, A098191, A098192, A098193, A098194, A098195.

Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &] - EulerPhi@ n, {n, 120}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d)) - eulerphi(n); \\ Michel Marcus, Feb 25 2014

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

a(n) > A063919(n) if n > 1.
a(A000040(k)) = 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) - 3/Pi^2 = 0.380252... . - Amiram Eldar, Aug 21 2023

Edited by R. J. Mathar, Mar 02 2009

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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A098188 Irregular triangle with 4 columns which contains in each row the members of a 4-cycle under the map x->A063919(x), iteration of summing the proper-unitary divisors.

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A098187 Initial seeds x which will enter a cycle of length 4 under the iteration of x -> A063919(x), the sum of proper unitary divisors.

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A332974 Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = usigma(k) - k is the sum of proper unitary divisors of k (A063919).

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A369895 Irregular triangle of iteration steps of A063919 until the end of the terminal cycle is reached, read by rows.

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A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

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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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A319902 Unitary sociable numbers of order 4.

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