A002191 -id:A002191 - cp's OEIS frontend

A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

1, 2, 3, 5, 4, 7, 6, 9, 13, 8, 10, 19, 14, 12, 29, 16, 21, 22, 37, 18, 27, 20, 43, 33, 34, 28, 49, 24, 61, 32, 67, 30, 73, 45, 57, 44, 40, 36, 50, 42, 52, 101, 63, 85, 109, 91, 74, 54

Offset: 1

N. J. A. Sloane

This is the least integer with the increasing sigma value A002191(n). For integers sorted on the ordered sigma values A007609(n), see A085790. - Lekraj Beedassy, Oct 08 2004

The sigma function (A000203) can't have a left nor a right inverse since it is neither injective nor surjective. The first column of the table A085790 (undefined when the row length A054973(n) = 0 <=> no x has sigma(x) = n) or A051444 (which has zeros filled in for these undefined values) are right-inverse of sigma on A002191 = range of sigma: one has A000203(A051444(n)) = A000203(A085790(n,1)) = n for all n in A002191 <=> A054973(n) > 0 <=> row A085790(n,.) nonempty <=> there is x with sigma(x) = n. Since sigma(6) = sigma(11) = 12, a hypothetical left inverse g must satisfy g(12) = 6 and g(12) = 11 which is impossible. Restricted to this list A002192 of smallest indices for the possible values of sigma, there exists a left inverse g such that g(sigma(x)) = x for all x in A002192. This equation defines the function g, i.e., g(A002191(n)) := a(n). A different left inverse exists on the set of largest pre-images for the possible values of sigma, {A085790(n,A054973(n)); n in A002191} = {1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, ...}. - M. F. Hasler, Nov 21 2019

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
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).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

A051444 is a better version of this sequence.

Cf. A000203, A002191, A007609, A007626, A054973, A085790.

m = 1000; Clear[f]; f[k_] := f[k] = Split[{DivisorSigma[1, #], #}& /@ Range[3k] // Sort, #1[[1]] == #2[[1]]&][[1 ;; m, 1]][[All, 2]]; f[k = m]; f[k = k+m]; While[f[k] != f[k, m], k = k+m]; A002192 = f[k] (* Jean-François Alcover, Oct 15 2015 *)

A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

Original entry on oeis.org

2, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 4, 4, 2, 1, 1, 4, 2, 1, 1, 2, 2, 4, 6, 2, 1, 3, 2, 1, 5, 4, 2, 4, 2, 4, 6, 1, 2, 3, 2, 4, 2, 4, 2, 2, 2, 6, 1, 3, 2, 1, 1, 4, 1, 5, 2, 4, 6, 2, 4, 2, 2, 2, 2, 4, 3, 3, 2, 4, 2, 1, 3, 6, 2, 1, 3, 2, 4, 6, 2, 4, 1, 5, 2, 4, 2, 4, 6, 2, 6, 4, 3, 1, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 6

Offset: 1

Labos Elemer, May 20 2003

nonn

			8 and 12 are the 6th and 7th possible values for sigma(x), since they are sum of divisors of x = 7 and x = 11 respectively, while 9, 10, 11 are impossible ones so 12 - 8 = 4 = a(6) = A002191(7) - A002191(6).
From _Michael De Vlieger_, Jul 22 2017: (Start)
First position of values:
Value   First position
    1         2
    2         1
    3        10
    4         6
    5        30
    6        24
    7       277
    8       165
    9       509
   10       150
   11       824
   12       400
   13     10970
   14      1400
   15     10448
   16      1182
   17     18731
   18      2218
   19    209237
   20      3420
   21    127385
   22      6910
   23     28899
   24      5377
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002191, A007609, A007369, A083532, A083533, A083534, A083535, A083536, A109323 (start of record gaps in A002191).

t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
(* Second program: *)
With[{nn = 300}, Differences@ TakeWhile[Union@ DivisorSigma[1, Range@ nn], # < nn &]] (* Michael De Vlieger, Jul 22 2017 *)

A061072 Smallest integer with A002191(n) divisors, i.e., the number of divisors equals the sum of the divisors of a different number.

Original entry on oeis.org

1, 4, 6, 12, 64, 24, 60, 4096, 192, 144, 180, 240, 360, 960, 720, 1073741824, 840, 1260, 786432, 36864, 1680, 2880, 15360, 2520, 6300, 6720, 2359296, 5040, 3221225472, 14400, 983040, 10080, 206158430208, 184320, 15120, 20160, 25200, 2985984, 9663676416, 27720

Offset: 1

Labos Elemer, May 28 2001

nonn

			For all values of sigma(x), i.e., of A002191, the smallest number with identical number of divisors is found at A005179(sigma(x)). E.g., 8 = A002191(6) is a possible divisor sum. The smallest number which has 8 divisors is 24 = A005179(8). See also comment to A008864, with special solutions of equation: sigma(x) = tau(y) = A000203(x) = A000005(y).

Amiram Eldar, Table of n, a(n) for n = 1..959

Cf. A000005, A000203, A002191, A005179, A008864.

A000005(a(n)) = A002191(n). I.e., if function A000005 is applied to this sequence, then values of A002191 are obtained. These terms are taken from A005179.

a(n) = A005179(A002191(n)). - David Wasserman, Jun 06 2002

More terms from David Wasserman, Jun 06 2002

Offset corrected by Sean A. Irvine, Jan 19 2023

A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

Original entry on oeis.org

1, 3, 7, 13, 31, 39, 91, 93, 127, 217, 307, 381, 403, 921, 961, 1093, 1209, 1651, 1723, 2149, 2801, 2821, 3279, 3541, 3937, 3991, 4953, 5113, 5169, 7651, 8011, 8191, 8403, 9517, 10303, 10623, 11811, 11973, 12061, 12493, 15339, 17293, 19531, 19607, 22399

Offset: 1

Michel Marcus, Jun 10 2014

nonn

Since 2 does not belong to A002191, all terms are odd.
All primes p that are in A023195 (Prime numbers that are the sum of the divisors of some n), are also in this sequence; and the prime factors of all terms can only belong to A023195.
Up to 10^7, only one term is a prime power: 961=31^2 (being a square, see A038688, A228061 and A243810).

			The divisors of 3 are 1 and 3 that both belong to A002191, 1 as sigma(1) and 3 as sigma(2).
The divisors of 39 are 1, 3, 13 and 39 all of which belong to A002191, 13 as sigma(9) 39 as sigma(18).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A000203, A002191, A023195.

Cf. A045572 (analog sequence with the sum of proper divisors instead).

N:= 10^6: # to get all terms up to N
A002191:= select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N):
A243765:= select(t -> numtheory[divisors](t) subset A002191, A002191); # Robert Israel, Jun 16 2014

list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
isok(n, lists) = {fordiv (n, d, if (!vecsearch(lists, d), return(0))); return(1);}
lista(nn) = {lists = list(nn); for(n=1, nn, if (isok(n, lists), print1(n, ", ")););}

A309727 a(n) is the least integer k such that for some iteration of sigma applied to k, one gets the n-th term of A002191, the list of possible values for the function sum of divisors.

Original entry on oeis.org

1, 2, 2, 5, 2, 2, 5, 9, 9, 2, 10, 19, 2, 5, 29, 16, 16, 22, 37, 10, 27, 19, 43, 33, 34, 5, 49, 2, 61, 16, 67, 29, 73, 45, 49, 43, 27, 22, 50, 19, 52, 101, 16, 85, 109, 22, 73, 5, 81, 33, 67, 64, 50, 86, 81, 137, 76, 66, 149, 111, 99, 157, 81, 106, 163, 2, 52, 173, 129

Offset: 1

Michel Marcus, Oct 14 2019

nonn

The set union of this sequence is 1 U A007369.

			For n = 5, A002191(5) is 7, and 4 iterations of sigma applied to 2 give 7, and no integer less than 2 will give 7, so a(5)=2.

Michel Marcus, Table of n, a(n) for n = 1..2503

Cf. A000203, A002191, A007369, A007497, A051572, A257349.

A257670 is a better version for this sequence.

list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
lista(nn) = {my(vs = list(nn), v = vector(#vs)); v[1] = 1; for (n=2, #vs, for (k=2, vs[n], my(kk=k); while (sigma(kk) <= vs[n], kk=sigma(kk)); if (kk == vs[n], v[n] = k; break););); v;}

a(n) = 2 when A002191(n) is in A007497.
a(n) = 5 when A002191(n) is in A051572.
a(n) = 16 when A002191(n) is in A257349.

A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, 47, 53, 39, 49, 59, 61, 32, 67, 71, 73, 45, 79, 83, 89, 36, 50, 77, 97, 101, 103, 107, 109, 91, 113, 95, 81, 75, 82, 64, 127, 131, 121, 137, 139, 119, 149, 151, 125

Offset: 1

M. F. Hasler, Nov 22 2019

nonn

			The possible values of sigma(x) are A002191 = {1, 3, 4, 6, 7, 8, 12, ...}.
The 7th value is 12 = sigma(x) for x = 6 or 11. Since 11 is the largest such value, a(7) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sigma), A002191 (range of sigma), A085790 (table of pre-images of x in A002191), A054973 (number of solutions of sigma(x) = n).

Cf. A051444 (smallest k such that sigma(k) = n).

A329821(n)=vecmax(invsigma(A002191(n))) \\ see Alekseyev link for invsigma(). An invsigmaMax() function is announced.

a(n) = A085790(m,A054973(m)) with m = A002191(n).

A054973 Number of numbers whose divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 0

Offset: 1

Henry Bottomley, May 16 2000

nonn

a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - Jaroslav Krizek, Mar 01 2010
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018
Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

			a(12) = 2 since 11 has factors 1 and 11 with 1 + 11 = 12 and 6 has factors 1, 2, 3 and 6 with 1 + 2 + 3 + 6 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sum-of-divisors function).
For partial sums see A074753.
Cf. A002191, A007609, A308039.
The non-strict version is A371284, ranks A371288.
These partitions have ranks A371283, unsorted version A275700.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
Cf. A001221, A002865, A008289, A371286, A371285.

nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* Robert G. Wilson v, May 14 2014 *)
Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* Gus Wiseman, Mar 22 2024 *)

a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ Michel Marcus, Oct 22 2013

a(n)=sum(k=1,n,sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013

first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ Charles R Greathouse IV, Mar 08 2017

A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308039. - Amiram Eldar, Dec 23 2024

Incorrect comment deleted by M. F. Hasler, Nov 21 2019

A007369 Numbers n such that sigma(x) = n has no solution.

Original entry on oeis.org

2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

With an initial 1, may be constructed inductively in stages from the list L = {1,2,3,....} by the following sieve procedure. Stage 1. Add 1 as the first term of the sequence a(n) and strike off 1 from L. Stage n+1. Add the first (i.e. leftmost) term k of L as a new term of the sequence a(n) and strike off k, sigma(k), sigma(sigma(k)),.... from L. - Joseph L. Pe, May 08 2002
This sieve is a special case of a more general sieve. Let D be a subset of N and let f be an injection on D satisfying f(n) > n. Define the sieve process as follows: 1. Start with the empty sequence S and let E = D. 2. Append the smallest element s of E to S. 3. Remove s, f(s), f(f(s)), f(f(f(s))), ... from E. 4. Go to step 2. After this sieving process, S = D - f(D). To get the current sequence, take f = sigma and D = {n | n >= 2}. - Max Alekseyev, Aug 08 2005
By analogy with the untouchable numbers (A005114), these numbers could be named "sigma-untouchable". - Daniel Lignon, Mar 28 2014
The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			a(4) = 10 because there is no x < 10 whose sigma(x) = 10.

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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].
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Complement of A002191.
See A083532 for the gaps, i.e., first differences.
See A048995 for the missed sums of nontrivial divisors.

a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ]

list(lim)=my(v=List(),u=vectorsmall(lim\1),t); for(n=1,lim, t=sigma(n); if(t<=lim, u[t]=1)); for(n=2,lim, if(u[n]==0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Mar 09 2017

A007369_list(LIM,m=0,L=List(),s)={for(n=2,LIM,(s=sigma(n-1))>LIM || bittest(m,s) || m+=1<M. F. Hasler, Mar 12 2018

A175192(a(n)) = 0, A054973(a(n)) = 0. - Jaroslav Krizek, Mar 01 2010

a(n) < 2n + sqrt(8n). - Charles R Greathouse IV, Oct 23 2015

More terms from David W. Wilson

A023195 Prime numbers that are the sum of the divisors of some n.

Original entry on oeis.org

3, 7, 13, 31, 127, 307, 1093, 1723, 2801, 3541, 5113, 8011, 8191, 10303, 17293, 19531, 28057, 30103, 30941, 86143, 88741, 131071, 147073, 292561, 459007, 492103, 524287, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 797161, 830833, 1191373

Offset: 1

David W. Wilson

nonn

If n > 2 and sigma(n) is prime, then n must be an even power of a prime number. For example, 1093 = sigma(3^6). - T. D. Noe, Jan 20 2004
All primes of the form 2^n-1 (Mersenne primes) are in the sequence because if n is a natural number then sigma(2^(n-1)) = 2^n-1. So A000668 is a subsequence of this sequence. If sigma(n) is prime then n is of the form p^(q-1) where both p & q are prime (the proof is easy). - Farideh Firoozbakht, May 28 2005
Primes of the form 1 + p + p^2 + ... + p^k where p is prime.
If n = sigma(p^k) is in the sequence, then k+1 is prime. - Franklin T. Adams-Watters, Dec 19 2011
Primes that are a repunit in a prime base. - Franklin T. Adams-Watters, Dec 19 2011.
Except for 3, these primes are particular Brazilian primes belonging to A085104. These prime numbers are also Brazilian primes of the form (p^x - 1)/(p^y - 1), p prime, belonging to A003424, with here x is prime, and y = 1. [See section V.4 of Quadrature article in Links.] - Bernard Schott, Dec 25 2012
From Bernard Schott, Dec 25 2012: (Start)
Others subsequences of this sequence:
  A053183 for 111_p = p^2 + p + 1 when p is prime.
  A190527 for 11111_p =  p^4 + p^3 + p^2 + p + 1 when p is prime.
  A194257 for 1111111_p = p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime. (End)
Subsequence of primes from A002191. - Michel Marcus, Jun 10 2014

			307 = 1 + 17 + 17^2; 307 and 17 are primes.

David W. Wilson, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Intersection of A002191 and A000040.

Cf. A000203, A000668, A023194 (the n that produce these primes), A053696, A085104, A003424, A053183, A190527, A194257.

t={3}; lim=10^9; n=1; While[p=Prime[n]; k=2; s=1+p+p^2; sHarvey P. Dale, Jun 18 2022 *)

upto(lim)=my(v=List([3]),t); forprime(p=2,solve(x=1,lim^(1/4), x^4+x^3+x^2+x+1-lim), forprime(e=5,1+log(lim)\log(p), if(isprime(t=sigma(p^(e-1))) && t<=lim, listput(v,t)))); forprime(p=2, solve(x=1,lim^(1/2),x^2+x+1-lim), if(isprime(t=p^2+p+1), listput(v,t))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Dec 20 2011

from sympy import isprime, divisor_sigma
A023195_list = sorted(set([3]+[n for n in (divisor_sigma(d**2) for d in range(1,10**4)) if isprime(n)])) # Chai Wah Wu, Jul 23 2016

A007368 Smallest k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002
Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006
According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006
Index of the first occurrence of n in A054973. - Jaroslav Krizek, Apr 25 2009

			a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)
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].
Kevin Ford, Sergei Konyagin, On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ, arXiv:1910.08452 [math.NT], 2019.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 166.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Cf. A000203, A054973, A002191, A007609.
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).
Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)

More terms from David W. Wilson

cp's OEIS Frontend

A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

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A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

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A061072 Smallest integer with A002191(n) divisors, i.e., the number of divisors equals the sum of the divisors of a different number.

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A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

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A309727 a(n) is the least integer k such that for some iteration of sigma applied to k, one gets the n-th term of A002191, the list of possible values for the function sum of divisors.

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A329821 Largest k for which sigma(k) = A002191(n), where A002191 = range of sigma, the sum-of-divisors function A000203.

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A054973 Number of numbers whose divisors sum to n.

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Mathematica

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A007369 Numbers n such that sigma(x) = n has no solution.

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