A005114 -id:A005114 - cp's OEIS frontend

A078923 Possible values of sigma(n)-n.

0, 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Benoit Cloitre, Dec 15 2002

nonn

To test whether k>1 is in the sequence, it suffices to check values of n up to (k-1)^2, since sigma(n)-n >= sqrt(n)+1 if n is composite.
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
The lower asymptotic density is at least 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 1 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number absent from this sequence.) - Charles R Greathouse IV, Dec 14 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link.

Cf. A000203, A001065, A002191, A007369. Complement of A005114.

lista(nn)=for (n=0, nn, if (n==1, kmax=2, kmax=(n-1)^2); for (k=1, kmax, if (sigma(k)-k == n, print1(n, ", "); break););); \\ Michel Marcus, Nov 11 2014

Edited by Dean Hickerson, Dec 19 2002

Offset fixed by Michel Marcus, Dec 19 2014

A115060 Maximum peak of aliquot sequence starting at n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 21, 19, 22, 21, 22, 23, 55, 25, 26, 27, 28, 29, 259, 31, 32, 33, 34, 35, 55, 37, 38, 39, 50, 41, 259, 43, 50, 45, 46, 47, 76, 49, 50, 51, 52, 53, 259, 55, 64, 57, 58, 59, 172, 61, 62, 63, 64, 65, 259

Offset: 1

Sergio Pimentel, Mar 06 2006

nonn

According to Catalan's conjecture all aliquot sequences end in a prime followed by 1, a perfect number, a friendly pair or an aliquot cycle. Some sequences seem to be open ended and keep growing forever i.e. 276. Most sequences only go down (i.e. 10 - 8 - 7 - 1), so for most cases in this sequence, a(n) = n. The first number to achieve a significantly high peak is 138

			a(24)=55 because the aliquot sequence starting at 24 is: 24 - 36 - 55 - 17 - 1 so the maximum peak of this sequence is 55.

Jinyuan Wang, Table of n, a(n) for n = 1..275
W. Creyaufmueller, Aliquot Sequences.
Paul Zimmerman, Aliquot Sequences.

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

from sympy import divisor_sigma as sigma
def aliquot(n):
    alst = []; seen = set(); i = n
    while i and i not in seen: alst.append(i); seen.add(i); i = sigma(i) - i
    return alst
def aupton(terms): return [max(aliquot(n)) for n in range(1, terms+1)]
print(aupton(66)) # Michael S. Branicky, Jul 11 2021

More terms from Jinyuan Wang, Jul 11 2021

A284156 4-untouchable numbers.

Original entry on oeis.org

298, 1006, 1016, 1108, 1204, 1492, 1502, 1940, 2164, 2344, 2370, 2770, 3116, 3358, 3410, 3482, 3596, 3676, 3688, 3976, 4076, 4164, 4354, 4870, 5206, 5634, 5770, 6104, 6206, 6332, 6488, 6696, 6850, 7008, 7118, 7290, 7496, 7586, 7654, 7812, 7922, 8164, 8396, 8434

Offset: 1

Anton Mosunov, Mar 21 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). 4-untouchable numbers are the numbers that lie in the image of s3(n), but not in the image of s4(n). Question: does the set of 4-untouchable numbers have a natural asymptotic density?

			All even numbers less than 298 have a preimage under s4(n), so they are not 4-untouchable.
a(1) = 298, because 298 = s3(668) but 668 is untouchable. Therefore 298 is not in the image of s4(n). Note that 668 is the only preimage of 298 under s3(n).
a(2) = 1006, because 1006 = s3(5366) but 5366 is untouchable.
a(3) = 1016, because 1016 = s3(4402) = s3(5378) but both 4402 and 5378 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.
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, A284147, A284187.

Several missing terms inserted by Jinyuan Wang, Jan 07 2025

A284187 5-untouchable numbers.

Original entry on oeis.org

838, 904, 1970, 2066, 2176, 3134, 3562, 4226, 4756, 5038, 5312, 5580, 5692, 6612, 6706, 7096, 7210, 7384, 9266, 9530, 9704, 10316, 10742, 10828, 11482, 11578, 11724, 12384, 12592, 12682, 13098, 13236, 13772, 14582, 14846, 15184, 15284, 15338, 15484, 15520, 15578

Offset: 1

Anton Mosunov, Mar 21 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). 5-untouchable numbers are the numbers that lie in the image of s4(n), but not in the image of s5(n). Question: does the set of 5-untouchable numbers have a natural asymptotic density?

			All even numbers less than 838 have a preimage under s5(n), so they are not 5-untouchable.
a(1) = 838, because 838 = s4(2588) but 2588 is untouchable. Therefore 838 is not in the image of s5(n). Note that 2588 is the only preimage of 838 under s4(n).
a(2) = 904, because 904 = s4(4402) = s4(5378) but both 4402 and 5378 are untouchable.
a(3) = 1970, because 1970 = s4(4312) but 4312 is 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.
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, A284147, A284156, A363461.

Several missing terms inserted by Jinyuan Wang, Jan 07 2025

A324277 Infinitary untouchable numbers: numbers that are not the sum of aliquot infinitary divisors of any number.

Original entry on oeis.org

2, 3, 4, 5, 38, 68, 80, 128, 158, 164, 188, 192, 206, 212, 224, 278, 290, 308, 326, 368, 380, 398, 416, 432, 458, 518, 530, 536, 542, 548, 578, 584, 600, 626, 632, 692, 702, 710, 752, 758, 770, 782, 788, 818, 822, 836, 852, 872, 896, 902, 926, 938, 968, 998

Offset: 1

Amiram Eldar, Feb 20 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2256 (terms below 30000)

Cf. A049417, A126168, A005114, A063948 (unitary), A324276 (bi-unitary), A324278 (exponential).

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := isigma[n] = Times @@ (fun @@@ FactorInteger[n]); untouchableQ[n_] := Catch[ Do[ If[n == isigma[k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 2, 1000}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

A306747 Untouchable numbers with a record gap to the next untouchable number.

Original entry on oeis.org

2, 5, 2642, 546046

Offset: 1

Amiram Eldar, Mar 07 2019

The record gap values are 3, 47, 62, 74.

Guy asks "How large can the gaps between untouchable numbers be?"

			5 is in the sequence since it is an untouchable number and the next untouchable number after it is 52 = 5 + 47 with a record gap of 47. The next gap which is larger than 47 occurs at 2642 which is followed by 2704 = 2642 + 62.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B10, p. 101.

Giovanni Resta, Untouchable numbers (terms up to 10^6)

Cf. A005114, A231965.

A324276 Bi-unitary untouchable numbers: numbers that are not the sum of aliquot bi-unitary divisors of any number.

Original entry on oeis.org

2, 3, 4, 5, 38, 68, 80, 96, 98, 128, 138, 146, 158, 164, 180, 188, 192, 206, 208, 210, 212, 222, 224, 248, 264, 278, 290, 300, 304, 308, 324, 326, 328, 338, 360, 374, 380, 390, 398, 416, 418, 420, 430, 432, 458, 476, 480, 488, 498, 516, 518, 530, 536, 542, 548

Offset: 1

Amiram Eldar, Feb 20 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3591 (terms below 30000)

Cf. A188999, A005114, A063948 (unitary), A324277 (infinitary), A324278 (exponential), A331970.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := bsigma[n] = Times @@ (fun @@@ FactorInteger[n]); untouchableQ[n_] := Catch[ Do[ If[n == bsigma[k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 2, 550}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

A048188 a(n) = phi(n-th untouchable number).

Original entry on oeis.org

1, 4, 24, 40, 32, 32, 60, 72, 54, 92, 102, 48, 72, 96, 80, 120, 130, 132, 88, 96, 112, 144, 144, 96, 132, 108, 162, 96, 108, 120, 168, 128, 140, 168, 192, 232, 156, 164, 168, 216, 192, 208, 144, 176, 276, 280, 192, 288, 192, 192, 312, 312, 276, 332, 264, 232, 192

Offset: 1

Naohiro Nomoto

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A005114, A048189, A048190, A048191.

a(n) = A000010(A005114(n)).

More terms from Amiram Eldar, Sep 23 2022

A048189 Let (u1,u2) be successive untouchable numbers such that phi(u1) = phi(u2); sequence gives values of u1.

Original entry on oeis.org

96, 292, 612, 626, 732, 1186, 1346, 1522, 1944, 2198, 3202, 3270, 3454, 3684, 3746, 4322, 5848, 5950, 6276, 6396, 6626, 7466, 7522, 7932, 7972, 8482, 8548, 8676, 12756, 12868, 13958, 15342, 15524, 15852, 15866, 17550, 17938, 18002, 18534, 19516, 20452, 22716

Offset: 1

Naohiro Nomoto

nonn

			96 is a term since 96 and 120 are successive untouchable numbers and phi(96) = phi(120).
292 is a term since 292 and 304 are successive untouchable numbers and phi(292) = phi(304).

Amiram Eldar, Table of n, a(n) for n = 1..4309 (terms 1..81 from Michel Marcus)

Cf. A000010, A005114, A048188, A048190, A048191.

Offset changed to 1 and more terms from Michel Marcus, Jun 08 2015

A048190 Let (u1,u2) be successive untouchable numbers such that phi(u1) = phi(u2); sequence gives values of u2.

Original entry on oeis.org

120, 304, 624, 628, 738, 1192, 1348, 1528, 1956, 2212, 3208, 3276, 3476, 3708, 3748, 4336, 5876, 5952, 6282, 6402, 6632, 7468, 7528, 7944, 7976, 8488, 8552, 8688, 12762, 12872, 13972, 15348, 15536, 15858, 15868, 17556, 17944, 18008, 18552, 19520, 20464, 22734

Offset: 1

Naohiro Nomoto

nonn

			120 is a term since 96 and 120 are successive untouchable numbers and phi(96) = phi(120).
304 is a term since 292 and 304 are successive untouchable numbers and phi(292) = phi(304).

Amiram Eldar, Table of n, a(n) for n = 1..4309 (terms 1..81 from Michel Marcus)

Cf. A000010, A005114, A048188, A048189, A048191.

Offset changed to 1 and more terms from Michel Marcus, Jun 08 2015

cp's OEIS Frontend

A078923 Possible values of sigma(n)-n.

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A115060 Maximum peak of aliquot sequence starting at n.

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A284156 4-untouchable numbers.

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A284187 5-untouchable numbers.

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A324277 Infinitary untouchable numbers: numbers that are not the sum of aliquot infinitary divisors of any number.

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A306747 Untouchable numbers with a record gap to the next untouchable number.

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A324276 Bi-unitary untouchable numbers: numbers that are not the sum of aliquot bi-unitary divisors of any number.

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A048188 a(n) = phi(n-th untouchable number).

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A048189 Let (u1,u2) be successive untouchable numbers such that phi(u1) = phi(u2); sequence gives values of u1.

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