A283152 -id:A283152 - cp's OEIS frontend

A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

Offset: 1

N. J. A. Sloane

Complement of A078923. - Lekraj Beedassy, Jul 19 2005
Chen & Zhao show that the lower density of this sequence is at least 0.06, improving on te Riele. - Charles R Greathouse IV, Dec 28 2013
Numbers k such that A048138(k) = 0. A048138(k) measures how "touchable" k is. - Jeppe Stig Nielsen, Jan 12 2020
From Amiram Eldar, Feb 13 2021: (Start)
The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
Erdős (1973) proved that the lower asymptotic density of untouchable numbers is positive, te Riele (1976) proved that it is > 0.0324, and Banks and Luca (2004, 2005) proved that it is > 1/48.
Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
The upper asymptotic density is less than 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 0 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number in this sequence.) - Charles R Greathouse IV, Dec 05 2022

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Giovanni Resta, Table of n, a(n) for n = 1..13863 (terms < 10^5; first 8153 terms from Klaus Brockhaus)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840.
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Experimental Mathematics (2018), pp. 1-12.
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 (in German).
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
Victor Meally, Letter to N. J. A. Sloane, no date.
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 272.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Carl Pomerance and Hee-Sung Yang, On untouchable numbers and related problems, 2012.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
Giovanni Resta, Untouchable numbers (the 150232 terms up to 10^6).
H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976. See chapter 9.
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A001065, A048138, A057709, A064000, A078923, A152454, A231964, A283152, A284147.

untouchableQ[n_] := Catch[ Do[ If[n == DivisorSigma[1, k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Print[n]; Sow[n]], {n, 2, 700}]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Benoit Cloitre *)

isA078923(n)=if(n==0 || n==1, return(1)); for(m=1,(n-1)^2, if( sigma(m)-m == n, return(1))); 0
isA005114(n)=!isA078923(n)
for(n=1,700, if (isA005114(n), print(n))) \\ R. J. Mathar, Aug 10 2006

is(n)=if(n%2 && n<4e18, return(n==5)); forfactored(m=1,(n-1)^2, if(sigma(m)-m[1]==n, return(0))); 1 \\ Charles R Greathouse IV, Dec 05 2022

from sympy import divisor_sigma as sigma
from functools import cache
@cache
def f(m): return sigma(m)-m
def okA005114(n):
    if n < 2: return 0
    return not any(f(m) == n for m in range(1, (n-1)**2+1))
print([k for k in range(289) if okA005114(k)]) # Michael S. Branicky, Nov 16 2024

# faster for intial segment of sequence
from itertools import count, islice
from sympy import divisor_sigma as sigma
def agen(): # generator of terms
    n, touchable, t = 2, {0, 1}, 1
    for m in count(2):
        touchable.add(sigma(m)-m)
        while m > t:
            if n not in touchable:
                yield n
            else:
                touchable.discard(n)
            n += 1
            t = (n-1)**2
print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 16 2024

More terms from David W. Wilson

A283156 Number of preimages of even integers under the sum-of-proper-divisors function.

Original entry on oeis.org

0, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 2, 0, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 1, 2, 4, 2, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 3, 2, 1, 3, 1, 1, 0, 2, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 2, 1, 1

Offset: 1

Anton Mosunov, Mar 01 2017

nonn

Let sigma(n) denote the sum of divisors function, and s(n):=sigma(n)-n. The k-th element a(k) corresponds to the number of solutions to 2k=s(m) in positive integers, where m is a variable. In 2016, C. Pomerance proved that, for every e > 0, the number of solutions is O_e((2k)^{2/3+e}).

Note that for odd numbers n the problem of solving n=s(m) is quite different from the case when n is even. According to a slightly stronger version of Goldbach's conjecture, for every odd number n there exist primes p and q such that n = s(pq) = p + q + 1. This conjecture was verified computationally by Oliveira e Silva to 10^18. Thus the problem is (almost) equivalent to counting the solutions to n=p+q+1 in primes.

			a(1)=0, because 2*1=s(m) has no solutions;
a(2)=1, because 2*2=s(9);
a(3)=2, because 2*3=s(6)=s(25);
a(4)=2, because 2*4=s(10)=s(49);
a(5)=1, because 2*5=s(14).

Anton Mosunov, Table of n, a(n) for n = 1..10000
R. K. Guy, J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107.
P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26.
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.
C. Pomerance, H.-S. Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.T. Oliveira e Silva, Goldbach conjecture verification, 2015.

Cf. A048138, A283152.

a(n) =  sum(k=1, (2*n-1)^2, (sigma(k) - k) == 2*n); \\ Michel Marcus, Mar 04 2017

a(n) = A048138(2*n). - 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 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 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

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.

cp's OEIS Frontend

A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

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A283156 Number of preimages of even integers under the sum-of-proper-divisors function.

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A284147 3-untouchable numbers.

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A284156 4-untouchable numbers.

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A284187 5-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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