A057709 -id:A057709 - cp's OEIS frontend

A357313 a(n) is the unique number m such that A001065(m) = A057709(n).

4, 9, 8, 15, 14, 21, 121, 289, 529, 46, 28, 841, 62, 24, 1369, 217, 2209, 106, 68, 3481, 54, 4489, 134, 5041, 66, 6241, 158, 6889, 166, 116, 9409, 10201, 218, 226, 148, 130, 114, 16129, 17161, 18769, 278, 90, 24649, 314, 26569, 27889, 108, 29929, 225, 32041, 244

Offset: 1

Amiram Eldar, Sep 23 2022

nonn

			a(1) = 4 since 4 is the unique solution m to A001065(m) = A057709(1) = 3.

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

Cf. A001065, A057709.

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

A001065(a(n)) = A057709(n).

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

Original entry on oeis.org

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

A048138 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is n.

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 2, 1, 5, 2, 3, 1, 3, 1, 4, 1, 1, 3, 4, 2, 5, 2, 3, 2, 3, 1, 6, 2, 4, 0, 3, 2, 6, 1, 5, 1, 3, 1, 6, 2, 3, 3, 6, 1, 6, 1, 2, 1, 5, 1, 8, 3, 4, 3, 5, 1, 7, 1, 6, 1, 4, 1, 8, 1, 5, 0, 5, 2, 9, 2, 4, 1, 4, 0, 9, 1, 3, 2, 6, 1, 8, 2, 7, 4

Offset: 2

Naohiro Nomoto

The offset is 2 since there are infinitely many numbers (all the primes) for which A001065 = 1.
The graph of this sequence, shifted by 1, looks similar to that of A061358, which counts Goldbach partitions of n. - T. D. Noe, Dec 05 2008
For n > 2, a(n) <= A000009(n) as all divisor lists must have distinct values. - Roderick MacPhee, Sep 13 2016
The smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k is A125601(n). - Bernard Schott, Mar 23 2023

			a(6) = 2 since 6 is the sum of the proper divisors of 6 and 25.

T. D. Noe, Table of n, a(n) for n = 2..10000
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.

Cf. A001065, A064440, A125601, A238895, A238896 (records).

Cf. A005114, A057709, A057710, A160133.

with(numtheory): for n from 2 to 150 do count := 0: for m from 1 to n^2 do if sigma(m) - m = n then count := count+1 fi: od: printf(`%d,`,count): od:

list(n)=my(v=vector(n-1),k); for(m=4,n^2, k=sigma(m)-m; if(k>1 & k<=n, v[k-1]++)); v \\ Charles R Greathouse IV, Apr 21 2011

From Bernard Schott, Mar 23 2023: (Start)
a(n) = 0 iff n is in A005114 (untouchable numbers).
a(n) = 1 iff n is in A057709 ("hermit" numbers).
a(n) = 2 iff n is in A057710.
a(n) > 1 iff n is in A160133. (End)

More terms from James Sellers, Feb 19 2001

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 *)

A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

Original entry on oeis.org

2, 3, 6, 21, 37, 31, 49, 79, 73, 91, 115, 127, 151, 121, 181, 169, 217, 265, 253, 271, 211, 301, 433, 379, 331, 361, 457, 391, 451, 655, 463, 541, 421, 775, 511, 769, 673, 715, 865, 691, 1015, 631, 1069, 1075, 721, 931, 781, 1123, 871, 925, 901, 1177, 991, 1297

Offset: 0

Klaus Brockhaus, Nov 27 2006

nonn

Minimal values for nodes of exact degree in aliquot sequences. Find each node's degree (number of predecessors) in aliquot sequences and choose the smallest value as the sequence member. - Ophir Spector, ospectoro (AT) yahoo.com Nov 25 2007

			a(4) = 37 since there are exactly four numbers (155, 203, 299, 323) whose sum of proper divisors is 37. For k < 37 there are either fewer or more numbers (32, 125, 161, 209, 221 for k = 31) whose sum of proper divisors is k.

Daniel Mondot, Table of n, a(n) for n = 0..5646 (First 157 terms from Ophir Spector. First 1000 terms from Ophir Spector and Donovan Johnson.)
W. Creyaufmueller, Aliquot sequences
Daniel Mondot, A-file, extended version of b-file but with some gaps towards the end.
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065, A048138, A070015, A123930, A080907, A115350, A121507, A037020, A126016, A057709, A057710, A063990.

{m=54;z=1500;y=600000;v=vector(z);for(n=2,y,s=sigma(n)-n; if(s

A357324 Numbers k such that there is a unique m for which the sum of the aliquot unitary divisors of m (A034460) is k.

Original entry on oeis.org

6, 9, 11, 13, 128, 150, 164, 222, 224, 332, 338, 390, 404, 416, 420, 458, 510, 548, 558, 570, 576, 582, 584, 598, 660, 668, 750, 788, 800, 810, 818, 822, 836, 852, 878, 884, 926, 930, 1046, 1118, 1200, 1202, 1230, 1244, 1250, 1260, 1284, 1298, 1304, 1382, 1422, 1472, 1478

Offset: 1

Amiram Eldar, Sep 24 2022

nonn

Numbers k such that A324938(k) = 1.

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

The unitary version of A057709.

Cf. A034460, A063948, A324938, A357325.

us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; m = 1500; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; Position[v, 1] // Flatten

a(n) = A034460(A357325(n)).

A135244 Largest 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

0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 28, 187, 841, 221, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 437, 1849, 403, 86, 493, 2209, 551, 94, 589, 0, 667, 2809, 713, 106, 703, 68, 697, 3481

Offset: 2

Ophir Spector (ospectoro(AT)yahoo.com), Nov 25 2007

nonn

Previous name: Aliquot predecessors with the largest values.
Find each node's predecessors in aliquot sequences and choose the largest predecessor.
Climb the aliquot trees on shortest paths (see A135245 = Climb the aliquot trees on thickest branches).
The sequence starts at offset 2, since all primes satisfy sigma(n)-n = 1. - Michel Marcus, Nov 11 2014

			a(25) = 143 since 25 has 3 predecessors (95,119,143), 143 being the largest.
a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Amiram Eldar, Table of n, a(n) for n = 2..10000 (terms 2..150 from Ophir Spector)
Wolfgang Creyaufmueller, Aliquot sequences.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence.

Cf. A001065, A005114, A125601, A135245, A057709, A057710, A063769, A080907, A121507, A037020, A126016.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]] = Max[s[[i]], n]], {n, 2, (max - 1)^2}]; Rest @ s]; seq[50]

lista(nn) = {for (n=2, nn, k = (n-1)^2; while(k && (sigma(k)-k != n), k--); print1(k, ", "););} \\ Michel Marcus, Nov 11 2014

a(1)=0 removed and offset set to 2 by Michel Marcus, Nov 11 2014

New name from Michel Marcus, Oct 31 2023

A357325 a(n) is the unique number m such that A034460(m) = A357324(n).

Original entry on oeis.org

6, 15, 21, 35, 250, 138, 4192, 10048, 6112, 748, 20736, 5968, 802, 12256, 41728, 3592, 498, 53632, 8656, 80128, 2284, 2308, 36352, 2372, 10288, 5272, 11728, 84352, 1594, 630, 6472, 48448, 6616, 50368, 1426, 1762, 102016, 172288, 32416, 8872, 2328, 9544, 19408

Offset: 1

Amiram Eldar, Sep 24 2022

nonn

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

The unitary version of A357313.

Cf. A034460, A057709, A357324.

us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; m = 1500; v = s = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++; s[[u]] = k], {k, 1, m^2}]; s[[Position[v, 1] // Flatten]]

A034460(a(n)) = A357324(n).

A372742 Numbers k such that there is a unique number m for which the sum of the aliquot coreful divisors of m (A336563) is k.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 24, 29, 31, 36, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 80, 83, 84, 89, 96, 97, 98, 101, 103, 107, 109, 112, 113, 127, 131, 135, 137, 139, 140, 149, 150, 151, 156, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198

Offset: 1

Amiram Eldar, May 12 2024

nonn

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
Numbers k such that A372739(k) = 1.
The corresponding values of m are in A372743.
Includes all prime numbers.

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

Cf. A336563, A372739, A372743.
A000040 is a subsequence.
Similar sequences: A057709, A357324, A361419.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, 1] // Flatten]; seq[200]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] -1) - 1) - n;}
lista(nmax) = {my(v = vector(nmax), i); for(k=1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 1, print1(k, ", ")));}

a(n) = A336563(A372743(n)).

A135245 Aliquot predecessors with the largest degrees.

Original entry on oeis.org

0, 0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 12, 55, 289, 65, 361, 91, 20, 85, 529, 143, 46, 133, 28, 187, 841, 161, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 341, 1849, 403, 86, 493, 2209, 551, 40, 481, 0, 667, 2809, 533, 106, 703, 68, 697, 3481

Offset: 1

Ophir Spector, ospectoro (AT) yahoo.com, Nov 25 2007

nonn

Find each node's predecessors in aliquot sequences and choose the node with largest number of predecessors.

Climb the aliquot trees on thickest branches (see A135244 = Climb the aliquot trees on shortest paths).

			a(25) = 143 since 25 has 3 predecessors (95,119,143) with degrees (4,5,7), 143 having the largest degree. a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Ophir Spector, Table of n, a(n) for n = 1..150
W. Creyaufmueller, Aliquot sequences
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065 A005114 A125601 A135244 A057709 A057710 A063769 A080907 A121507 A037020 A126016.

cp's OEIS Frontend

A357313 a(n) is the unique number m such that A001065(m) = A057709(n).

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

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A048138 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is n.

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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).

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A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

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A357324 Numbers k such that there is a unique m for which the sum of the aliquot unitary divisors of m (A034460) is k.

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A135244 Largest 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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