A048138 -id:A048138 - cp's OEIS frontend

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

N. J. A. Sloane, R. K. Guy

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 91.

T. D. Noe, Table of n, a(n) for n = 1..10000
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].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
P. Pollack and C. 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 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

numlib::sigma(n)-n$ n=1..81 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
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
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

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

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

Original entry on oeis.org

4, 9, 6, 25, 8, 10, 49, 15, 14, 21, 121, 27, 35, 22, 169, 16, 33, 12, 26, 39, 55, 289, 65, 77, 34, 361, 18, 51, 91, 20, 38, 57, 85, 529, 95, 119, 143, 46, 69, 133, 28, 115, 187, 841, 32, 125, 161, 209, 221, 58, 961, 45, 87, 247, 62, 93, 145, 253, 24, 155, 203, 299, 323, 1369

Offset: 2

T. D. Noe, Dec 05 2008

In an aliquot sequence, all numbers in row n can be predecessors of n. This sequence is a permutation of the composite numbers; number k appears in row A001065(k). We start with n=2 because every prime would be in row 1. Note that row 2 is empty -- as are all the rows listed in A005114. Row n contains A048138(n) numbers. When n is prime, the largest number in row n+1 is n^2. When n>7 is odd, the largest number in row n is less than ((n-1)/2)^2 and (if a strong form of the Goldbach conjecture is true) has the form pq, with primes p

In row n, the first term is A070015(n), and the last term is A135244(n). - Michel Marcus, Nov 11 2014

The first row with several terms is row(6), where the difference between extreme terms is 25-6=19. The next row with a smaller difference is row(13) with a difference 35-27=8. And the next one is row(454) with a difference 602-596=6. Is there a next row with a smaller difference? - Michel Marcus, Nov 11 2014

			Irregular triangle starts:
  ; (empty row at n=2)
  4;
  9;
  ; (empty row at n=5)
  6, 25;
  8;
  10, 49;
  15;
  14;
  21;
  121;
  27, 35;
  22, 169;
  16, 33;
  12, 26;
  39, 55;
  289;
  ...

T. D. Noe, Rows n=2..1000 of triangle, flattened
Michel Marcus, Rows n=2..1000 of triangle, not flattened

Cf. A001065, A005114, A048138.

Cf. A070015, A135244, A135245.

N:= 100: # for rows 2 to N, flattened
for s from 2 to N do B[s]:= NULL od:
for k from 1 to N^2 do
  if not isprime(k) then
    s:= numtheory:-sigma(k)-k;
    if s <= N then
       B[s]:= B[s],k;
    fi
  fi
od:
seq(B[s],s=2..N); # Robert Israel, Nov 11 2014

nn=100; s=Table[{},{nn}]; Do[k=DivisorSigma[1,n]-n; If[1

row(n) = select(x->(sigma(x)-x)==n, [1..n^2]); \\ Michel Marcus, Feb 25 2025

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]]

A057709 Numbers k such that there is a unique m for which the sum of the aliquot parts of m (A001065) is k.

Original entry on oeis.org

3, 4, 7, 9, 10, 11, 12, 18, 24, 26, 28, 30, 34, 36, 38, 39, 48, 56, 58, 60, 66, 68, 70, 72, 78, 80, 82, 84, 86, 94, 98, 102, 112, 116, 118, 122, 126, 128, 132, 138, 142, 144, 158, 160, 164, 168, 172, 174, 178, 180, 190, 192, 204, 208, 212, 220, 222, 224, 228, 250

Offset: 1

Jack Brennen, Oct 24 2000

nonn

Alanen (1972) used the term "hermit" for a number k such that x = k is the only solution to A001065(x) = k. These numbers are the perfect numbers (A000396) in this sequence. Of the first 4 perfect numbers, 6, 28, 496 and 8128, only 28 is a term. - Amiram Eldar, Mar 03 2021

			12 is a member of the sequence because s(121)=12 (and because no other integer m satisfies s(m) = 12).
18 is included because the sum of aliquot parts of 289 = 1+17 = 18, this being the only number with this property. 6 is not included because the sum of aliquot parts of 6 = 1+2+3 = 6 and the sum of aliquot parts of 25 = 1+5 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
Eric Weisstein's World of Mathematics, Restricted Divisor Function.
Eric Weisstein's World of Mathematics, Aliquot Sequence.

Cf. A000396, A001065, A005114, A048138, A057710.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2 }]; Position[s, 1] // Flatten]; seq[250] (* Amiram Eldar, Dec 26 2020 *)

Removed 1 from the sequence. - T. D. Noe, Dec 02 2008

A238896 Number of times that the number A238895(n) occurs in the sum of proper divisors function (A001065).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 13, 15, 20, 21, 24, 25, 27, 32, 34, 41, 44, 46, 56, 59, 70, 74, 76, 86, 92, 99, 115, 116, 129, 138, 145, 156, 164, 169, 171, 193, 199, 222, 223, 242, 246, 268, 270, 278, 295, 304, 334, 343, 365, 397, 398, 433, 452, 471, 482, 521

Offset: 1

T. D. Noe, Mar 10 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..245

Cf. A001065, A005114, A048138, A238895.

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

a(n) = A048138(A238895(n)). - Amiram Eldar, Sep 23 2022

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

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A064440 Numbers k such that A048138(k) is a prime and sets a new record for such primes.

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A064465 Numbers k such that A048138(k) = A034444(k).

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A104886 Sequence A048138, the number of times that a positive integer occurs as the sum of proper divisors, if Goldbach partitions (two odd primes, which account for most of the values) are ignored.

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A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than 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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A152454 Irregular triangle in which row n lists the numbers whose proper divisors sum to n.

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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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