A284187 -id:A284187 - cp's OEIS frontend

A283152 2-untouchable numbers.

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

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

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

A363461 Least n-untouchable number.

Original entry on oeis.org

2, 208, 388, 298, 838

Offset: 1

Jinyuan Wang, Jun 03 2023

Let s^m(k) denote the m-th iterate of s(k) = sigma(k) - k. n-untouchable numbers are the numbers that lie in the image of s^(n-1)(k), but not in the image of s^n(k).

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.

Cf. A005114, A152454, A283152, A284147, A284156, A284187.

A363875 Numbers k such that there is no odd number whose aliquot sequence contains k.

Original entry on oeis.org

2, 28, 52, 88, 96, 120, 124, 146, 162, 188, 206, 208, 210, 216, 238, 246, 248, 250, 262, 268, 276, 288, 290, 292, 298, 304, 306, 322, 324, 326, 336, 342, 362, 372, 388, 396, 406, 408, 412, 426, 428, 430, 438, 448, 452, 472, 474, 478, 486, 494, 498, 508, 516

Offset: 1

Jinyuan Wang, Jun 25 2023

nonn

k is in sequence iff k can never be reached when iterating the map x -> A001065(x) starting with any odd number m.
Assuming the stronger version of Goldbach conjecture, iff k is in the sequence, there are infinitely many odd numbers whose aliquot sequence contain k.
Supersequence of A005114 (except 5), A283152, A284147, A284156, A284187, ..., and untouchable perfect numbers (28, 137438691328, ...), untouchable amicable numbers (A238382), untouchable sociable numbers.

Andrew R. Booker, Finite connected components of the aliquot graph, arXiv:1610.07471 [math.NT], 2017.
Jean-Luc Garambois, Le graphe infini des suites aliquotes (in French).
Wikipedia, Aliquot sequence

Cf. A001065, A005114, A238382, A283152, A284147, A284156, A284187.

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A283152 2-untouchable numbers.

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A284147 3-untouchable numbers.

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A284156 4-untouchable numbers.

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A363461 Least n-untouchable number.

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A363875 Numbers k such that there is no odd number whose aliquot sequence contains k.

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