A002827 -id:A002827 - cp's OEIS frontend

A127654 Unitary aspiring numbers.

66, 78, 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, 3096, 3140, 3172

Offset: 1

Ant King, Jan 24 2007

nonn

A unitary aspiring number is an integer whose unitary aliquot sequences ends by meeting a unitary-perfect number (A098185) in its trajectory, but is not unitary-perfect itself. There are 1693 such numbers <=100000 and of these 82860 and 97020 generate the longest unitary aliquot sequences (according to A097032), each having length 18 and ending with the unitary perfect number 90.

			a(5) = 476 because the fifth non-unitary-perfect number whose unitary aliquot sequence ends in a unitary-perfect number is 476.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Herman J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, Amsterdam, 1972.
Herman J. J. te Riele, Further Results on Unitary Aliquot Sequences, NW 2/73, Mathematisch Centrum, Amsterdam, 1973.

Cf. A002827, A097032, A127652, A097010, A098185, A127653, A063991, 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]]; UnitaryPerfectNumberQ[0] = 0; UnitaryPerfectNumberQ[k_] := If[sstar[k] == k, True, False]; UnitaryAspiringNumberQ[k_] := If[UnitaryPerfectNumberQ[Last[ UnitaryTrajectory[k]]] && ! UnitaryPerfectNumberQ[k], True, False]; Select[Range[2500], UnitaryAspiringNumberQ[ # ] &]
s[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[3200], q] (* Amiram Eldar, Mar 11 2023 *)

More terms from Amiram Eldar, Mar 11 2023

A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 3, 2, 4, 1, 3, 1, 2, 1, 3, 1, 0, 1, 1, 3, 4, 2, 4, 1, 5, 2, 4, 1, 0, 1, 2, 3, 3, 1, 4, 1, 4, 3, 4, 1, 0, 2, 2, 2, 2, 1, 0, 1, 5, 2, 1, 2, 2, 1, 5, 2, 6, 1, 4, 1, 5, 2, 4, 2, 1, 1, 5, 1, 3, 1, 5, 2, 4, 4, 4, 1, 0, 3, 4, 3, 5, 2, 5, 1, 5, 3, 1, 1, 1, 1, 5, 5

Offset: 1

Antti Karttunen, Sep 22 2018, after Labos Elemer's A097033

nonn

This sequence implements the original definition given for A097033.

			For n = 1, A063919(1) = 1, that is, we immediately end with a terminal cycle (of length 1 in this case), thus there are no transient part, and a(1) = 0.
For n = 2, A063919(2) = 1, and A063919(1) = 1, so we end with a terminal cycle after a transient part of length 1, thus a(2) = 1.
For n = 30, A063919(30) = 42, A063919(42) = 54, A063919(54) = 30, thus a(30) = a(42) = a(54) = 0, as 30, 42 and 54 are all contained in their own terminal cycle, 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.
If a(n) = 0, then n is a term in an attractor set like A002827, A063991, A097024, A097030.

Antti Karttunen, Table of n, a(n) for n = 1..87360

Cf. A003062, A063919, A097031, A097033, A318880, A318882.

a063919[1] = 1; (* function a[] in A063919 by Jean-François Alcover *)
a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]]-n/;n>1
transient[k_] := Module[{iter=NestWhileList[a063919, k, UnsameQ, All]}, Position[iter, Last[iter]][[1, 1]]]-1
a318883[n_] := Map[transient, Range[n]]
a318883[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));
A318883(n) = { my(visited = Map()); for(j=1, oo, if(mapisdefined(visited, n), return(mapget(visited, n)-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); };
A318883(n) = { my(visited = List([]), k); for(j=1, oo, if((k = pil(n,visited)) > 0, return(k-1)); listput(visited, n); n = A063919(n)); };

a(n) = A318882(n) - A097031(n).

a(n) = A097033(n) + A318880(n) - 1.

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

Original entry on oeis.org

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)));

A097034 Initial values for iteration of the function f(x) = A063919(x) such that the iteration ends in a 14-cycle, i.e., in A097030.

Original entry on oeis.org

1506, 1518, 1806, 1902, 1914, 1938, 1950, 2226, 2382, 2394, 2406, 2418, 2478, 2826, 2910, 2946, 2958, 3234, 3282, 3294, 3330, 3510, 3522, 3534, 3546, 3582, 3642, 3654, 3774, 3906, 3954, 3966, 3978, 4146, 4158, 4194, 4434, 4446, 4854, 4866, 4878, 5262

Offset: 1

Labos Elemer, Aug 30 2004

nonn

			n=1506 is here because its iteration list = {1506, 1518, 1938, 2382, 2394, 2406, [2418, ...., 3582, 2418}. After a transient of length 6, the iteration ends in a cycle of length 14.

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
a097034Q[k_] :=
 Module[{iter = NestWhileList[a063919, k, UnsameQ, All]},
  Apply[Subtract, Reverse[Flatten[Position[iter, Last[iter]], 1]]] ==
   14]
a097034[n_] := Select[Range[n], a097034Q]
a097034[5262] (* Hartmut F. W. Hoft, Jan 25 2024 *)

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.

Original entry on oeis.org

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 *)

A322486 Semi-unitary perfect numbers: numbers k such that susigma(k) = 2k, where susigma(k) is the sum of the semi-unitary divisors of k (A322485).

Original entry on oeis.org

6, 60, 90, 264, 3960, 4560, 8736, 13770, 131040, 384384, 605880, 5765760, 20049120, 882161280, 23253135360

Offset: 1

Amiram Eldar, Dec 11 2018

a(16) <= 1846273228800. - David A. Corneth, Dec 11 2018

			264 is in the sequence since its sum of semi-unitary divisors is susigma(264) = 528 = 2 * 264.

Cf. A000396, A002827, A322485.

f[p_, e_] := (p^Floor[(e+1)/2] - 1)/(p-1) + p^e; susigma[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; aQ[n_] := susigma[n]==2n; Select[Range[10000], aQ]

ssu(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k,1], e=f[k,2]); f[k,1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k,2] = 1;); factorback(f);} \\ A322485
isok(n) = ssu(n) == 2*n; \\ Michel Marcus, Dec 14 2018

A335202 Unitary Zumkeller numbers (A290466) whose set of unitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

6, 60, 70, 90, 3230, 3770, 4030, 4510, 5170, 5390, 5830, 50388, 87360, 269990, 442365, 544310, 592670, 740870, 1341230, 1772870, 4173070, 4199030, 5719266, 5728842, 5743206, 34473582, 624032630, 812851182, 1109686930, 1113445430, 2280959890, 55157757606

Offset: 1

Amiram Eldar, May 26 2020

nonn

			60 is a term since there is only one partition of its set of unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, into 2 disjoint sets whose sum is equal: 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.

Giovanni Resta, Table of n, a(n) for n = 1..48 (terms < 10^12)

The unitary version of A083209.
Subsequence of A290466.
A002827 is a subsequence.

uzQ[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]]; Select[Range[6000], uzQ]

Terms a(19) and beyond from Giovanni Resta, May 30 2020

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)));

A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1791, 2047, 2431, 4095, 8191, 14335, 14847, 16383, 27391, 32767, 44031, 57855, 65535, 114687, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 8978431, 12058623, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A002827, A004767, A034448.

For similar sequences, see A336700, A387410, A387415, A387419.

A000265(n) = (n>>valuation(n,2));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
isA387418(n) = !((1+A000265(A034448(n)))%A000265(1+n));

cp's OEIS Frontend

A127654 Unitary aspiring numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A318883 Number of transient terms if unitary-proper-divisor-sum-function f(x) = A063919(x) is iterated and the initial value is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A097034 Initial values for iteration of the function f(x) = A063919(x) such that the iteration ends in a 14-cycle, i.e., in A097030.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A322486 Semi-unitary perfect numbers: numbers k such that susigma(k) = 2k, where susigma(k) is the sum of the semi-unitary divisors of k (A322485).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A335202 Unitary Zumkeller numbers (A290466) whose set of unitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A319917 Unitary sociable numbers of order six.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI