A005114 -id:A005114 - cp's OEIS frontend

A057709 Numbers k such that there is a unique m for which the sum of the aliquot parts of m (A001065) is k.

3, 4, 7, 9, 10, 11, 12, 18, 24, 26, 28, 30, 34, 36, 38, 39, 48, 56, 58, 60, 66, 68, 70, 72, 78, 80, 82, 84, 86, 94, 98, 102, 112, 116, 118, 122, 126, 128, 132, 138, 142, 144, 158, 160, 164, 168, 172, 174, 178, 180, 190, 192, 204, 208, 212, 220, 222, 224, 228, 250

Offset: 1

Jack Brennen, Oct 24 2000

nonn

Alanen (1972) used the term "hermit" for a number k such that x = k is the only solution to A001065(x) = k. These numbers are the perfect numbers (A000396) in this sequence. Of the first 4 perfect numbers, 6, 28, 496 and 8128, only 28 is a term. - Amiram Eldar, Mar 03 2021

			12 is a member of the sequence because s(121)=12 (and because no other integer m satisfies s(m) = 12).
18 is included because the sum of aliquot parts of 289 = 1+17 = 18, this being the only number with this property. 6 is not included because the sum of aliquot parts of 6 = 1+2+3 = 6 and the sum of aliquot parts of 25 = 1+5 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
Eric Weisstein's World of Mathematics, Restricted Divisor Function.
Eric Weisstein's World of Mathematics, Aliquot Sequence.

Cf. A000396, A001065, A005114, A048138, A057710.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2 }]; Position[s, 1] // Flatten]; seq[250] (* Amiram Eldar, Dec 26 2020 *)

Removed 1 from the sequence. - T. D. Noe, Dec 02 2008

A115350 Termination of the aliquot sequence starting at n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, 11, 3, 53, 3, 17, 41, 23, 31, 59, 43, 61, 7, 41, 41, 19, 3, 67, 31, 13, 43, 71, 3, 73, 43, 7, 41, 19, 3, 79, 41, 43, 43

Offset: 1

Sergio Pimentel, Mar 07 2006

nonn

Catalan's conjecture [not yet established and probably false] is that every aliquot sequence terminates in a prime number followed by 1, a perfect number, a friendly pair or an aliquot cycle.

a(n) = the prime number if the sequence terminates in a prime followed by 1, a(n) = a perfect number if the sequence terminates in a perfect number, a(n) = the smallest number of the cycle if the sequence terminates in an aliquot cycle, a(n) = 0 if the sequence is open ended. So far 276 is the smallest number for which the termination of the aliquot sequence is not known.

			a(12)=3 since the aliquot sequence starting at 12 is: 12 - 16 - 15 - 9 - 4 - 3.
a(95)=6 since the aliquot sequence starting at 95 is: 95 - 25 - 6 - 6 ...

W. Creyaufmueller, Aliquot Sequences.
R. J. Mathar, Table of n, a(n) for n= 1,...,12572 with -1 substituted for a(n) where terminations are not yet known.
R. J. Mathar, Illustration of Aliquot Sequence Mergers (2014)
Paul Zimmerman, Aliquot Sequences.

Cf. A098007, A098008, A098009, A098010, A003023, A044050, A007906, A037020, A063769, A005114.

a[n_] := If[n == 1, 1, FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, n] /. {k__, 1, 0, 0} :> {k} // Last];
Array[a, 100] (* Jean-François Alcover, Mar 28 2020 *)

Edited by N. J. A. Sloane, Aug 14 2006

a(61)-a(80) from R. J. Mathar's list by Robert Price, Mar 16 2019

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

Original entry on oeis.org

1, 12, 24, 72, 168, 240, 336, 360, 504, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 15120, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800

Offset: 1

Douglas E. Iannucci, Oct 22 2008

nonn

Sequence A206027 has the number of solutions.

			sigma(m)=1 has only one solution: m=1.
sigma(m)=12 has two solutions, m=6 and m=11; 12 is the smallest number with more than one such solutions.
sigma(m)=24 has three solutions, m=14,m=15 and m=23; 24 is the smallest number with more than two such solutions.
sigma(m)=72 has five solutions, m=30, m=46, m=51, m=55 and m=71; 72 is the smallest number with more than three such solutions.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000203 (sum of divisors of n), A054973 (number of numbers whose divisors sum to n), A007368 (smallest k such that sigma(x) = k has exactly n solutions).
Cf. A206027.
Cf. Untouchable numbers (A005114), sigma-untouchable numbers (A007369) and highly touchable numbers (A238895).

t = DivisorSigma[1, Range[10^6]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)

{m=3650000; v=vectorsmall(m); for(n=1, m, s=sigma(n); if(s<=m, v[s]++)); g=0; j=1; while(j<=m, if(v[j]<=g, j++, g=v[j]; print1(j, ",")))} \\ Klaus Brockhaus, Oct 27 2008

Extended beyond a(15) by Klaus Brockhaus, Oct 27 2008

A048995 Numbers that are not the sum of the nontrivial factors (excluding 1 and n) of some natural number.

Original entry on oeis.org

1, 4, 51, 87, 95, 119, 123, 145, 161, 187, 205, 209, 215, 237, 245, 247, 261, 267, 275, 287, 289, 291, 303, 305, 321, 323, 325, 335, 341, 371, 405, 407, 425, 429, 447, 471, 473, 497, 515, 517, 519, 529, 539, 551, 555, 561, 575, 583, 611, 623, 625, 627, 657

Offset: 1

Bill Taylor (W.Taylor(AT)math.canterbury.ac.nz)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Complement of A048050.

Cf. A005114 (the same property with the sum of proper divisors) and A007369 (the same property with the sum of all divisors).

a048995[n_Integer] := Module[{t = Table[i, {i, n}], a048050, k},
  a048050[m_] := Plus @@ Take[Divisors[m], {2, -2}];
  Do[
   If[a048050[k] == 0 || a048050[k] > n, Null, t[[a048050[k]]] = 0],
   {k, 2, n^2}];
  Drop[DeleteDuplicates[t], {2}]
]; a048995[660] (* Michael De Vlieger, Nov 30 2014 *)

mx=8479; v=vector(mx); for(i=2, mx^2, ch=sigma(i)-i-1; if(ch<=mx, if(ch>0, v[ch]=1))); c=0; for(i=1, mx, if(v[i]==0, c++; write("b048995.txt", c " " i))) /* Donovan Johnson, Feb 11 2013 */

Corrected and extended by Jud McCranie

A238896 Number of times that the number A238895(n) occurs in the sum of proper divisors function (A001065).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 13, 15, 20, 21, 24, 25, 27, 32, 34, 41, 44, 46, 56, 59, 70, 74, 76, 86, 92, 99, 115, 116, 129, 138, 145, 156, 164, 169, 171, 193, 199, 222, 223, 242, 246, 268, 270, 278, 295, 304, 334, 343, 365, 397, 398, 433, 452, 471, 482, 521

Offset: 1

T. D. Noe, Mar 10 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..245

Cf. A001065, A005114, A048138, A238895.

nn = 1000; s = Table[0, {nn}]; Do[k = DivisorSigma[1, n] - n; If[0 < k <= nn, s[[k]]++], {n, nn^2}]; mx = -1; t = {}; Do[If[s[[n]] > mx, mx = s[[n]]; AppendTo[t, {n, mx}]], {n, 2, nn}]; Transpose[t][[2]]

a(n) = A048138(A238895(n)). - Amiram Eldar, Sep 23 2022

A283152 2-untouchable numbers.

Original entry on oeis.org

208, 250, 362, 396, 412, 428, 438, 452, 478, 486, 494, 508, 672, 712, 716, 772, 844, 900, 906, 950, 1042, 1048, 1086, 1090, 1112, 1132, 1140, 1252, 1262, 1310, 1338, 1372, 1518, 1548, 1574, 1590, 1592, 1644, 1676, 1678, 1686, 1752, 1756, 1796, 1808, 1810, 1854

Offset: 1

Anton Mosunov, Mar 01 2017

nonn

Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.
For n > 1, let s2(n) := s(s(n)). 2-untouchable numbers are the numbers that lie in the image of s(n), but not in the image of s2(n). Question: does the set of 2-untouchable numbers have a natural asymptotic density?
Let U(X) denote the total number of 2-untouchable numbers up to X. Then
  U(10^4) = 368
  U(10^5) = 4143
  U(10^6) = 46854
  U(10^7) = 508197
  U(10^8) = 5348219
  U(2*10^8) = 14616451

			All even numbers less than 208 have a preimage under s2(n), so they are not 2-untouchable.
a(1) = 208, because 208 = s(268) but 268 is untouchable. Therefore 208 is not in the image of s2(n). Note that 268 is the only preimage of 208 under s(n).
a(2) = 250, because 250 = s(290) but 290 is untouchable.
a(3) = 362, because 362 = s(430) = s(718) but both 430 and 718 are untouchable.

Anton Mosunov, Table of n, a(n) for n = 1..10000
K. Chum, R. K. Guy, and M. J. Jacobson, Numerical and statistics analysis of aliquot sequences, Exp. Math. 29 (2020) 414-425, Table 4.
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
Carl Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Cf. A005114, A152454, A284147, A284156, A284187.

preim(n) =  my(v = []); for (k=1, (n-1)^2, if (sigma(k)-k == n, v = concat(v, k))); v;
isunt(n) = if (n==1, 1, for (k=1, (n-1)^2, if (sigma(k)-k == n, return(0))); 1);
isok(n) =  v = preim(n); if (#v, b = 1; for (k=1, #v, b = b && isunt(v[k])); b, 0); \\ Michel Marcus, Mar 04 2017

A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.

Original entry on oeis.org

2, 4, 6, 40, 106, 314, 1954, 2234, 2794, 11194, 22394, 58234, 111994, 160154, 291194, 425594, 560554, 1022554, 1455994, 1601594, 3203194, 11703994, 16743994, 21781754, 24751994, 53253194, 60860794, 79587194, 95295194, 181060874, 287123194, 435635194, 973772794

Offset: 1

Anton Mosunov, Mar 01 2017

nonn

Let sigma(n) denote the sum of divisors function, and s(n):=sigma(n)-n. Let r(n) denote the number of solutions to n=s(m) and put a(1):=2. a(2) is equal to the smallest number such that r(a(2)) > r(a(1)). a(3) is equal to the smallest number such that r(a(3)) > r(a(2)), and so on.
Pomerance proved that, for every e > 0, the number of solutions to n = s(m) when n is even is O_e(n^{2/3+e}).
There are 49 elements in this sequence which do not exceed 2^40. The largest element, 690100611194, has 139 preimages.

			a(1)=2, because 2=s(m) has 0 solutions;
a(2)=4, because 4=s(9);
a(3)=6, because 6=s(6)=s(25);
a(4)=40, because 40=s(44)=s(74)=s(81);
a(5)=106, because 106=s(80)=s(104)=s(110)=s(206);
a(6)=314, because 314=s(370)=s(406)=s(442)=s(622)=s(313^2);
a(7)=1954, because 1954=s(1856)=s(1952)=s(2216)=s(2702)=s(3014)=s(3902);
a(8)=2234, because 2234=s(2536)=s(2770)=s(3454)=s(3562)=s(3706)=s(3886)=s(3922);
a(9)=2794, because 2794=s(3176)=s(3716)=s(3470)=s(3878)=s(4334)=s(4658)=s(4958)=s(4982)=s(5582).

Anton Mosunov, Table of n, a(n) for n = 1..49
Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, Numerical and Statistical Analysis of Aliquot Sequences, arXiv:2110.14136 [math.NT], 2021.
C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Cf. A005114, A283156, A287233, A287238, A287247, A287251, A287262.

v=vectorsmall(10^8);
for(n=2,#v,t=(sigma(n)-n)/2;if(denominator(t)==1 && t<=#v, v[t]++))
r=0;for(n=1,#v, if(v[n]>r,r=v[n];print1(2*n", "))) \\ Charles R Greathouse IV, Mar 02 2017

a(20)-a(25) from Charles R Greathouse IV, Mar 02 2017
a(26)-a(31) from Anton Mosunov, Mar 03 2017
a(32)-a(49) from Anton Mosunov, Apr 20 2017

A284147 3-untouchable numbers.

Original entry on oeis.org

388, 606, 696, 790, 918, 1264, 1330, 1344, 1350, 1468, 1480, 1496, 1634, 1688, 1800, 1938, 1966, 1990, 2006, 2026, 2102, 2122, 2202, 2220, 2318, 2402, 2456, 2538, 2780, 2830, 2916, 2962, 2966, 2998, 3224, 3544, 3806, 3926, 4208, 4292, 4330, 4404, 4446, 4466

Offset: 1

Anton Mosunov, Mar 20 2017

nonn

Let sigma(n) denote the sum of divisors of n, and s(n) := sigma(n) - n, with the convention that s(0)=0. Untouchable numbers are those numbers that do not lie in the image of s(n), and they were studied extensively (see the references). In 2016, Pollack and Pomerance conjectured that the set of untouchable numbers has a natural asymptotic density.

Let sk(n) denote the k-th iterate of s(n). 3-untouchable numbers are the numbers that lie in the image of s2(n), but not in the image of s3(n). Question: does the set of 3-untouchable numbers have a natural asymptotic density?

			All even numbers less than 388 have a preimage under s3(n), so they are not 2-untouchable.
a(1) = 388, because 388 = s2(668) but 668 is untouchable. Therefore 388 is not in the image of s3(n). Note that 668 is the only preimage of 388 under s2(n).
a(2) = 606, because 606 = s2(474) but 474 is untouchable.
a(3) = 696, because 696 = s2(276) = s2(306) but both 276 and 306 are untouchable.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, Numerical and Statistical Analysis of Aliquot Sequences, arXiv:2110.14136 [math.NT], 2021. and Exp. Math. 29 (2020) 414-425
R. K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
Carl Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.

Cf. A005114, A152454, A283152, A284156, A284187.

Several missing terms inserted by Jinyuan Wang, Jan 05 2025

A057710 Positive integers k with exactly 2 aliquot sequence predecessors. In other words, there are exactly two solutions x for which s(x) = n. The function s(x) here is the sum of all proper divisors of x (A001065).

Original entry on oeis.org

6, 8, 13, 14, 15, 16, 17, 19, 20, 22, 23, 27, 29, 32, 42, 44, 46, 50, 54, 62, 69, 90, 92, 100, 104, 108, 110, 114, 130, 136, 148, 150, 152, 156, 166, 170, 176, 182, 184, 186, 198, 200, 202, 214, 230, 232, 234, 236, 240, 242, 244, 254, 258, 266, 272, 280, 286

Offset: 1

Jack Brennen, Oct 24 2000

nonn

			14 is a member of the sequence because s(22) = 14 and s(169) = 14 (and because no other integer x satisfies s(x) = 14).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Restricted Divisor Function.
Eric Weisstein's World of Mathematics, Aliquot sequence.

Cf. A001065, A005114, A057709.

len = max = 57; f[List] := (s = Select[ Split[ Sort[ Table[ DivisorSigma[1, n] - n, {n, 1, max *= 2}]]], Length[#] == 2 & ][[All, 1]]; s [[1 ;; Min[len, Length[s]]]]); FixedPoint[f, {}] (* _Jean-François Alcover, Oct 07 2011 *)

cp's OEIS Frontend

A057709 Numbers k such that there is a unique m for which the sum of the aliquot parts of m (A001065) is k.

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A115350 Termination of the aliquot sequence starting at n.

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A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

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A048995 Numbers that are not the sum of the nontrivial factors (excluding 1 and n) of some natural number.

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A238896 Number of times that the number A238895(n) occurs in the sum of proper divisors function (A001065).

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A283152 2-untouchable numbers.

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A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.

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