A152454 -id:A152454 - 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

A085790 Integers sorted by the sum of their divisors.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 9, 13, 8, 10, 17, 19, 14, 15, 23, 12, 29, 16, 25, 21, 31, 22, 37, 18, 27, 20, 26, 41, 43, 33, 35, 47, 34, 53, 28, 39, 49, 24, 38, 59, 61, 32, 67, 30, 46, 51, 55, 71, 73, 45, 57, 79, 44, 65, 83, 40, 58, 89, 36, 50, 42, 62, 69, 77, 52, 97, 101, 63, 103, 85

Offset: 1

Hugo Pfoertner, Jul 23 2003

Integers having the same sum of divisors are sorted in ascending order, e.g., sigma(14)=sigma(15)=sigma(23)=24 -> a(15)=14, a(16)=15, a(17)=23.
Also an irregular triangle where the k-th row consists of all numbers with divisor sum k. See A054973(k) for the k-th row length. - Jeppe Stig Nielsen, Jan 29 2015
By definition this is a permutation of the positive integers. Also positive integers of A299762. - Omar E. Pol, Mar 14 2018

			a(9) = 9, a(10) = 13, a(11) = 8 because sigma(9) = 9 + 3 + 1 = 13, sigma(13) = 13 + 1 = 14, sigma(8) = 8 + 4 + 2 + 1 = 15 and there are no other numbers with those sigma values.
Irregular triangle starts: (row numbers to the left are not part of the sequence)
   n : row(n)
   1 : 1,
   2 :
   3 : 2,
   4 : 3,
   5 :
   6 : 5,
   7 : 4,
   8 : 7,
   9 :
  10 :
  11 :
  12 : 6, 11,
  13 : 9,
  14 : 13,
  15 : 8,
  16 :
  17 :
  18 : 10, 17,
  19 :
  20 : 19,
  21 :
  22 :
  23 :
  24 : 14, 15, 23,
  25 :
- _Jeppe Stig Nielsen_, Feb 02 2015, edited by _M. F. Hasler_, Nov 21 2019

Hugo Pfoertner, Table of n, a(n) for n=1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005
Jeppe Stig Nielsen, First 10000 rows of the triangle for a(n)
Index entries for sequences that are permutations of the natural numbers

Cf. A000203 (sigma), A007609 (values taken by sigma, with multiplicity), A002191 (possible values for sigma), A002192 (first column).

Cf. A152454 (similar sequence for proper divisors only (aliquot parts)).

SortBy[Table[{n,DivisorSigma[1,n]},{n,120}],Last][[;;,1]] (* Harvey P. Dale, Sep 10 2024 *)

A085790_row(n)=invsigma(n) \\ Cf. Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

A238895 Numbers m > 1 such that a record number of numbers k have m as the sum of the proper divisors of k.

Original entry on oeis.org

2, 3, 6, 21, 31, 49, 73, 91, 115, 121, 169, 211, 301, 331, 361, 391, 421, 511, 631, 721, 781, 841, 1051, 1261, 1471, 1561, 1681, 1891, 2101, 2311, 2521, 2731, 3151, 3361, 3571, 3991, 4201, 4411, 4621, 5251, 5461, 6091, 6511, 6721, 6931, 7771, 7981, 8191, 9031

Offset: 1

T. D. Noe, Mar 10 2014

nonn

The number of times that a(n) appears in A001065 is A238896(n).
By analogy with the untouchable numbers (A005114) and the highly composite numbers (A002182), these numbers can be named "highly touchable" (see Lignon). - Daniel Lignon, Mar 21 2014
Indices of record values in A048138. - Franklin T. Adams-Watters, Jul 27 2014

			For 2, there are no numbers.
For 3, there is 1 number: 4.
For 6, there are 2 numbers: 6 and 25.
For 21, there are 3 numbers: 18, 51, 91.
For 31, there are 5 numbers: 32, 125, 161, 209, 221.
For 49, there are 6 numbers: 75, 215, 287, 407, 527, 551.

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 317 (in French).

Amiram Eldar, Table of n, a(n) for n = 1..245 (terms 1..139 from Daniel Mondot)

Cf. A001065, A005114, A048138, A238896.
Cf. A152454 (row n lists the numbers whose proper divisors sum to n).
Cf. A239625 (irregular table giving the rows of numbers that produce a(n)).

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

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

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

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

A253303 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to 2*n, when there are at least two such x's.

Original entry on oeis.org

16, 136, 186, 1352, 1340, 1356, 1414, 1276, 1026, 15640, 6742, 1968, 35786, 924, 11076, 11812, 61714, 14556, 76862, 6440, 12774, 70444, 62446, 16080, 24904, 16152, 27900, 65464, 36026, 41100, 85810, 56008, 50142, 23152

Offset: 1

Michel Marcus, Dec 30 2014

Only integers m that satisfy A048138(m) > 1 are considered here.

			The integers whose sum of proper divisors is 16 are 12 and 26, and gcd(12, 26) is 2, so a(1) = 16 (see Example section of A152454).

Cf. A001065 (sum of proper divisors), A048138, A152454, A253302.

build(nb) = {vs = vector(nb); nc = nb^2; forcomposite(n=2, nc, val = sigma(n)-n; if (val <= nb, if (vs[val] == 0, vs[val] = -n, vs[val] = gcd(vs[val], n)););); vs[1] = 1; vs;}
lista() = {vd = build(100000); vmax = 80; vr = vector(vmax); forstep (ig=2, vmax, 2, for (i=1, #vd, w = vd[i]; if (w ==ig, vr[ig] = i; break;););); forstep (i=2, #vr, 2, if (vr[i] == 0, break, print1(vr[i], ", ")));}

a(n) = A253302(2*n).

A252997 Numbers n such that sigma(x) - x = n has at least two solutions, with each x having the same squarefree kernel, where sigma(x) is the sum of divisor function (A000203).

Original entry on oeis.org

218, 189648, 720240, 119967120, 129705984, 517941905, 707902440, 1321744320, 98890370304, 99080219520, 119922568640, 139834382688, 347612467648, 580542318720, 952717920000, 1064902900320, 1153644808680, 2255573174400, 3903820736256, 6859688278905, 10944640212480, 14424196864000

Offset: 1

Naohiro Nomoto, Dec 25 2014

nonn

Numbers n such that n = A001065(j) = A001065(k) and A007947(j) = A007947(k), where j != k.

			218 is the sum of proper divisors of 250 and 160, and rad(250) = rad(160) = 10, hence 218 is in the sequence with j=250 and k=160.
Other examples of n and j, k:
For n = 189648, j = 95832, k = 85536.
For n = 720240, j = 288120, k = 246960.
For n = 119967120, j = 38755080, k = 34398000.
For n = 129705984, j = 71614464, k = 60424704.

Carlos Rivera, Prime Puzzle 774. S(i) and Rad(i), The Prime Puzzles and Problems Connection.

Cf. A001065 (sum of proper divisors of n), A007947 (squarefree kernel of n).

Cf. A048138, A152454, A254035.

a(6) onward from Fred Schneider, Feb 07 2015

A366110 a(n) is the difference between the maximum and minimum number whose proper divisors sum to n, or 0 if there is no such number.

Original entry on oeis.org

0, 0, 0, 0, 19, 0, 39, 0, 0, 0, 0, 8, 147, 17, 14, 16, 0, 12, 327, 73, 18, 28, 0, 48, 0, 64, 0, 72, 0, 189, 903, 202, 0, 160, 0, 168, 0, 0, 37, 328, 1651, 387, 1767, 280, 34, 364, 0, 476, 54, 448, 0, 432, 2767, 677, 0, 604, 0, 432, 0, 528, 3603, 753, 66, 826, 0, 768, 0, 720, 0

Offset: 2

Michel Marcus, Oct 28 2023

nonn

A152454 is the irregular triangle in which row n lists the numbers whose proper divisors sum to n.

			A152454 begins as []; [4]; [9]; []; [6, 25]; [8]; [10, 49]...
so sequence begins 0, 0, 0, 0, 19, 0, 39, ...

Michel Marcus, Table of n, a(n) for n = 2..10001

Cf. A001065, A152454.
Cf. A070015, A135244.
Cf. A005114, A057709.

lista(nn) = my(v = vector(nn, k, [])); forcomposite (i=1, nn^2, my(x=sigma(i)-i); if (x <=  nn, v[x] = concat(v[x], i));); vector(nn-1, k, k++; if (#v[k], vecmax(v[k]) - vecmin(v[k])));

a(n) = A135244(n) - A070015(n).

a(A005114(n)) = a(A057709(n)) = 0.

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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PARI

PARI

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A085790 Integers sorted by the sum of their divisors.

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A238895 Numbers m > 1 such that a record number of numbers k have m as the sum of the proper divisors of k.

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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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A284187 5-untouchable numbers.

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A253303 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to 2*n, when there are at least two such x's.

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