A063991 -id:A063991 - cp's OEIS frontend

A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.

25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116, 119, 134, 142, 143, 154, 162, 170, 175, 182, 189, 190, 200, 220, 226, 242, 245, 262, 273

Offset: 1

Ant King, Jan 24 2007

Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.

			a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.

Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot Sequences.

Cf. A127161, A127162, A127163, A127164, A063769, A063990, A097032, A098007, A097010, A127653, A098185, A127654, A063991, A127655, A097037, A097036.

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]

Sequence gives those values of n for which A097032(n)>A098007(n).

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

A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

			Because A034460(6) = 6, a(6) = 1.
Because A034460(30) = 42, A034460(42) = 54, A034460(54) = 30, a(30) = a(42) = a(54) = 3.
Because A034460(90) = 90, a(90) = 1. Because A034460(78) = 90, a(78) = 0, as even though 78 ends into a cycle of one, it itself is not a part of that cycle.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A002827 (positions of ones), A063991 (of 2's), A319902 (of 4's), A097024 (of 5's), A319917 (of 6's), A319937 (of 10's), A097030 (of 14's), A327157 (of all nonzero terms).

Cf. also A034448, A034460, A097031, A318880, A318882, A327162.

a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1 &]] - n /; n > 0
cycleL[k_] := Module[{nL=NestWhileList[a034460, k, UnsameQ, All]}, If[k==Last[nL], Length[nL]-1, 0]]
a327159[n_] := Map[cycleL, Range[n]]
a327159[120] (* Hartmut F. W. Hoft, Feb 04 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A327159(n,orgn=n,xs=Set([])) = if(1==n,0,if(vecsearch(xs,n), if(n==orgn,length(xs),0), xs = setunion([n],xs); A327159(A034460(n),orgn,xs)));

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]

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);

A292019 List of pairs of unitary amicable numbers (m, n) with record value of m/n.

Original entry on oeis.org

114, 126, 44772, 49308, 241110, 242730, 10254970, 10273670, 766292835, 766512285, 17454440640, 17454615360

Offset: 1

Amiram Eldar, Sep 07 2017

The 2 members in each pair are adjacent to each other in the sequence.

The unitary version of A287026.

			The ratios m/n are
114/126 = 0.90476...
44772/49308 = 0.90800...
241110/242730 = 0.99332...
10254970/10273670 = 0.99818...
766292835/766512285 = 0.99971...
17454440640/17454615360 = 0.99999...

Cf. A002952, A002953, A063991, A287026.

A292020 List of pairs of unitary amicable numbers (m, n) with record low values of m/n.

Original entry on oeis.org

114, 126, 18018, 22302, 32130, 40446, 197340, 286500, 703972667580, 1057831128900

Offset: 1

Amiram Eldar, Sep 07 2017

The 2 members in each pair are adjacent to each other in the sequence.

The unitary version of A287011.

			The ratios m/n are
114/126 = 0.904...
18018/22302 = 0.807...
32130/40446 = 0.794...
197340/286500 = 0.688...
703972667580/1057831128900 = 0.665...

Cf. A002952, A002953, A063991, A287011.

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A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.

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A319917 Unitary sociable numbers of order six.

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A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

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A319937 Unitary sociable numbers of order 10.

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A292019 List of pairs of unitary amicable numbers (m, n) with record value of m/n.

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A292020 List of pairs of unitary amicable numbers (m, n) with record low values of m/n.

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