A319902 -id:A319902 - cp's OEIS frontend

A327157 Numbers that are members of unitary sigma aliquot cycles (union of unitary perfect, unitary amicable and unitary sociable numbers).

6, 30, 42, 54, 60, 90, 114, 126, 1140, 1260, 1482, 1878, 1890, 2142, 2178, 2418, 2958, 3522, 3534, 3582, 3774, 3906, 3954, 3966, 3978, 4146, 4158, 4434, 4446, 18018, 22302, 24180, 29580, 32130, 35220, 35238, 35340, 35820, 37740, 38682, 39060, 39540, 39660, 39780, 40446, 41460, 41580, 44340, 44460, 44772, 45402

Offset: 1

Antti Karttunen, Sep 17 2019

nonn

Positions of nonzeros in A327159.
Numbers n for which n = A034460^k(n) for some k >= 1, where A034460^k(n) means k-fold application of A034460 starting from n.
The terms that are not multiples of 6 are: 142310, 168730, 1077890, 1099390, 1156870, 1292570, ..., that seem all to be present in A063991.
Among the first 440 terms, there are numbers present in 1-cycles (A002827), 2-cycles (A063991), and also cycles of sizes 3, 4 (A319902), 5 (A097024), 6 (A319917), 14 (A097030), 25, 26, 39 and 65.

			6 is a member as A034460(6) = 6.
30 is a member as A034460(A034460(A034460(30))) = 30.

Antti Karttunen, Table of n, a(n) for n = 1..440
J. O. M. Pedersen, Known Unitary Sociable Numbers of order different from four [Via Internet Archive Wayback-Machine]

Cf. A002827, A063991, A097024, A097030, A319902, A319917, A319937 (subsequences), A034448, A034460, A097031, A327159.

Subsequence of A003062.

(* Function cycleL[] and support a034460[] are defined in A327159 *)
a327157[n_] := Map[cycleL, Range[n]]
a327157[45402] (* Hartmut F. W. Hoft, Feb 04 2024 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A034460(n) = (A034448(n) - n);
memo327159 = Map();
A327159(n) = if(1==n,0,my(v,orgn=n,xs=Set([])); if(mapisdefined(memo327159, n, &v), v, while(n && !vecsearch(xs,n), xs = setunion([n],xs); n = A034460(n); if(mapisdefined(memo327159,n),for(i=1,#xs,mapput(memo327159,xs[i],0)); return(0))); if(n==orgn,v = length(xs); for(i=1,v,mapput(memo327159,xs[i],v)), v = 0; mapput(memo327159,orgn,v)); (v)));
k=0; n=0; while(k<=1001, n++; if(t=A327159(n), k++; print(n," -> ",t); write("b327157.txt", k," ", 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)));

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

A319915 Smallest member of bi-unitary sociable quadruples.

Original entry on oeis.org

162, 1026, 1620, 10098, 10260, 41800, 51282, 100980, 107920, 512820, 1479006, 4612720, 4938136, 14790060, 14800240, 23168840, 28158165, 32440716, 55204500, 81128632, 84392560, 88886448, 209524210, 283604220, 325903500, 498215416, 572062304, 881697520

Offset: 1

Amiram Eldar, Oct 01 2018

nonn

The bi-unitary version of A090615.

			162 is in the sequence since the iterations of the sum of bi-unitary proper divisors function (A188999(n) - n) are cyclic with a period of 4: 162, 174, 186, 198, 162, ... and 162 is the smallest member of the quadruple.

Cf. A090615, A188999, A292980, A292981, A319902.

fun[p_, e_]:=If[Mod[e, 2]==1, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)];
bs[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]-n;seq[n_]:=NestList [bs, n,4][[2;;5]] ;aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s,n]==1]; Do[If[aQ[n],Print[n]],{n,1,10^9}]

fn(n) = {if (n==1, 1, f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f) - n;);}
isok(n) = my(v = vector(5)); v[1] = n; for(k=2, 5, v[k] = fn(v[k-1])); (v[5] == n) && (vecmin(v) == n) && (#vecsort(v,,8)==4); \\ Michel Marcus, Oct 02 2018

is(n) = my(c = n); for(i = 1, 3, c = fn(c); if(c <= n, return(0))); c = fn(c); c == n \\ uses Michel Marcus' fn David A. Corneth, Oct 02 2018

A323331 Smallest member of sociable quadruples using Dedekind psi function (A001615).

Original entry on oeis.org

11398670, 22797340, 38369450, 45594680, 56993350, 59334310, 76738900, 91189360, 113986700, 118668620, 153477800, 182378720, 209524210, 227973400, 237337240, 268586150, 284966750, 306955600, 364757440, 419048420, 455946800, 474674480, 537172300, 539867650, 569933500

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

Numbers k whose iterations of k -> A001615(k) - k are cyclic with a period of 4, and in each cyclic quadruple k is the least of the 4 members.

			11398670 is in the sequence since the iterations of k -> A001615(k) - k are cyclic with a period of 4: 11398670, 11475730, 12474350, 14093650, 11398670, ... and 11398670 is the smallest member of the quadruple.

Cf. A001615, A090615, A319902, A319915, A323327, A323328, A323329, A323330.

t[0]=0; t[1]=0; t[n_]:=(Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n;
seq[n_]:=NestList [t, n, 4][[2;; 5]] ; aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s, n]==1]; s={}; Do[If[aQ[n], AppendTo[s, n]], {n, 1, 10^9}]; s

A361811 Smallest members of infinitary sociable quadruples.

Original entry on oeis.org

1026, 10098, 10260, 41800, 45696, 100980, 241824, 685440, 4938136, 13959680, 14958944, 25581600, 28158165, 32440716, 36072320, 55204500, 74062944, 81128632, 149589440, 178327008, 192793770, 209524210, 283604220, 319848642, 498215416, 581112000, 740629440, 1236402232

Offset: 1

Amiram Eldar, Mar 25 2023

nonn

The first 8 terms were found by Cohen (1990).

			1026 is a term since the iterations of the sum of aliquot infinitary divisors function (A126168) that start with 1026 are cyclic with period 4: 1026, 1374, 1386, 1494, 1026, ..., and 1026 is the smallest member of the quadruple.
The first five quadruples are {1026, 1374, 1386, 1494}, {10098, 15822, 19458, 15102}, {10260, 13740, 13860, 14940}, {41800, 51800, 66760, 83540}, {45696, 101184, 94656, 88944}.

Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Jan Munch Pedersen, Known Infinitary Sociable Numbers of order four.

Cf. A049417, A126168.
Cf. A007357 (period 1), A126169 and A126170 (period 2).
Subsequence of A004607 (all cycles of length > 2).
Similar sequences: A090615 (all divisors), A319902 (unitary), A319915 (bi-unitary).

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]]>0, 1 + p^(2^(m-j)), 1], {j, 1, m}]]; infs[n_] := Times @@ f @@@ FactorInteger[n] - n;  infs[1] = 0; seq[n_] := NestList[infs, n, 4][[2;; 5]] ; q[n_] := Module[{s = seq[n]}, n == Min[s] && Count[s, n] == 1]; Select[Range[10^6], q]

infs(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
is(n) = {my(m = n); for(k = 1, 4, m = infs(m); if(k < 4 && m <= n, return(0))); m == n; }

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A327157 Numbers that are members of unitary sigma aliquot cycles (union of unitary perfect, unitary amicable and unitary sociable numbers).

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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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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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A319915 Smallest member of bi-unitary sociable quadruples.

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A323331 Smallest member of sociable quadruples using Dedekind psi function (A001615).

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A361811 Smallest members of infinitary sociable quadruples.

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