A319917 -id:A319917 - 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)));

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)));

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);

cp's OEIS Frontend

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

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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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