A002191 -id:A002191 - cp's OEIS frontend

A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

1, 3, 4, 6, 7, 8, 12, 12, 13, 14, 15, 18, 18, 20, 24, 24, 24, 28, 30, 31, 31, 32, 32, 36, 38, 39, 40, 42, 42, 42, 44, 48, 48, 48, 54, 54, 56, 56, 57, 60, 60, 60, 62, 63, 68, 72, 72, 72, 72, 72, 74, 78, 80, 80, 84, 84, 84, 90, 90, 90, 91, 93, 96, 96, 96, 96, 98, 98

Offset: 1

Walter Nissen

A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010

a(n) is the median of the values of A000203(m) from m=1 to m=2n-1. (This needs confirmation as it relies on the assumption that A000203(n) is always bigger than the median of the values A000203(x) from x=1 to x=n.) - Chayim Lowen, May 27 2015

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A002191 (duplicates removed), A007368, A085790.

sort(select(`<=`,map(numtheory:-sigma,[$1..1000]),1001)); # Robert Israel, Jun 04 2015

terms = 68; ClearAll[t]; t[k_] := t[k] = Sort[ Table[ DivisorSigma[1, n], {n, 1, k*terms}]][[1 ;; terms]]; t[k = 2]; While[t[k] != t[k-1], k++]; t[k] (* Jean-François Alcover, Nov 21 2012 *)
With[{nn=80},Take[Sort[DivisorSigma[1,Range[nn*100]]],nn]] (* Harvey P. Dale, Mar 09 2016 *)

list(lim)=select(k->k<=lim,Set(apply(sigma,[1..lim\1]))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A085790(n)). - Jinyuan Wang, Apr 15 2020

A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 9, 13, 8, 0, 0, 10, 0, 19, 0, 0, 0, 14, 0, 0, 0, 12, 0, 29, 16, 21, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 0, 43, 0, 0, 0, 33, 0, 0, 0, 0, 0, 34, 0, 28, 49, 0, 0, 24, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 0, 73, 0, 0, 0, 45, 0, 57, 0, 0, 0, 44, 0, 0, 0, 0, 0

Offset: 1

Jud McCranie

Column 1 of A299762. - Omar E. Pol, Mar 14 2018

This is a right inverse of sigma = A000203 on A002191 = range(sigma): if n is in A002191, then there is some x with sigma(x) = n, and by definition a(n) is the smallest such x, so sigma(a(n)) = n. - M. F. Hasler, Nov 22 2019

			sigma(1) = 1, so a(1) = 1.
There is no k with sigma(k) = 2, since sigma(k) >= k + 1 for all k > 1 and sigma(1) = 1, so a(2) = 0.
sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.

R. K. Guy, Unsolved Problems Theory of Numbers, B1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.

Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]

a(n)=for(k=1,n,if(sigma(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Mar 09 2014

A051444(n)=if(n=invsigma(n),vecmin(n)) \\ See Alekseyev link for invsigma(). An update including invsigmaMin = A051444 is planned. - M. F. Hasler, Nov 21 2019

Edited by M. F. Hasler, Nov 22 2019

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

A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

Original entry on oeis.org

12, 18, 24, 31, 32, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 98, 104, 108, 114, 120, 124, 126, 128, 132, 140, 144, 152, 156, 168, 180, 182, 186, 192, 210, 216, 224, 228, 234, 240, 248, 252, 264, 270, 272, 280, 288, 294, 308, 312, 320, 324, 336, 342, 360, 372, 378, 384, 390

Offset: 1

Jaroslav Krizek, Apr 25 2009

nonn

Numbers k with A054973(k) >= 2. Numbers k which occur in A000203 more than once.
Numbers k = A007609(n) with A007609(n+1) - A007609(n) = 0.
Does this sequence have finite density? - Franklin T. Adams-Watters, Jun 18 2009
See A300869 for the odd terms, much less frequent since they can only occur for x = k^2 or 2*k^2. - M. F. Hasler, Mar 16 2018

			a(1) = 12 as the multiplicity of the value 12 is 2: 12 = sigma(6) = sigma(11).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1095 from Franklin T. Adams-Watters)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A007609, A007368.
Subsequence of A002191.
Odd terms are listed in A300869.

na(n) = local(v, s); v=vector(n);for(k=1,n,s=sigma(k);if(s<=n,v[s]++));v
la(n) = local(v, r); v=na(n);r=[];for(k=1,n,if(v[k]>1,r=concat(r,[k])));r \\ Franklin T. Adams-Watters, Jun 18 2009

is(k) = invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

Edited and extended by R. J. Mathar, Apr 28 2009

A175192 a(n) = characteristic function of numbers k such that A000203(m) = k has solution, where A000203(m) = sums of divisors of m.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Jaroslav Krizek, Mar 01 2010

nonn

a(n) = 1 if A000203(m) = n for some m, else 0.

a(n) = 1 for n such that A054973(n) >= 1. a(n) = 0 for n such that A054973(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Characteristic function of A002191.

Cf. A000203, A054973.

nn = 200; t = Union[Select[DivisorSigma[1, Range[nn]], # <= nn &]]; t2 = Table[0, {200}]; t2[[t]] = 1; t2 (* T. D. Noe, Jan 24 2012 *)

up_to = 65537
v175192 = vector(up_to);
for(k=1, up_to, t=sigma(k); if(t<=up_to, v175192[t] = 1)); \\ See also code in A054973.
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,v175192,"b175192_upto65537.txt");
\\ Antti Karttunen, Oct 20 2017

More terms from Antti Karttunen, Oct 20 2017

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 11, 9, 13, 8, 0, 0, 10, 17, 0, 19, 0, 0, 0, 14, 15, 23, 0, 0, 0, 12, 0, 29, 16, 25, 21, 31, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 26, 41, 0, 43, 0, 0, 0, 33, 35, 47, 0, 0, 0, 0, 0, 34, 53, 0, 28, 39, 49, 0, 0, 24, 38, 59, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 46, 51, 55, 71, 0, 73

Offset: 1

Omar E. Pol, Mar 12 2018

Essentially the same as the triangle described in the example section of A085790, but with 0's added in empty rows.

Are the records the same as A008578?

			First 24 rows of triangle T(n,k):
-----------------------
. n / k:  1   2   3 ...
-----------------------
| 1|      1;
| 2|      0;
| 3|      2;
| 4|      3;
| 5|      0;
| 6|      5;
| 7|      4;
| 8|      7;
| 9|      0;
|10|      0;
|11|      0;
|12|      6, 11;
|13|      9;
|14|     13;
|15|      8;
|16|      0;
|17|      0;
|18|     10, 17;
|19|      0;
|20|     19;
|21|      0;
|22|      0;
|23|      0;
|24|     14, 15, 23;
...
For n = 23 there are no positive integers whose sum of divisors is 23, so T(23, 1) = 0, which is the only element in the 23rd row of the triangle.
For n = 24 there are three positive integers whose sum of divisors is 24; they are 14, 15 and 23, since sigma(14) = 1 + 2 + 7 + 14 = 24, sigma(15) = 1 + 3 + 5 + 15 = 24 and sigma(23) = 1 + 23 = 24, so the 24th row of the triangle is [14, 15, 23].

Row sums give A258913.
Column 1 gives A051444.
Right border gives A057637.
Positive terms give A085790.
Row n has A054973(n) positive integers.
Positive terms in the first column give A002192.
Indices of the rows that contain a zero give A007369.
Indices of the rows that contain positive terms give A002191.
Cf. A000203, A007368, A007609, A008578.

With[{nn = 74}, ReplacePart[ConstantArray[{0}, nn], PositionIndex@ Array[DivisorSigma[1, #] &, nn]]] // Flatten (* Michael De Vlieger, Mar 16 2018 *)

sigma(T(n,k)) = n, if T(n,k) >= 1.

A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

Original entry on oeis.org

1, 1, 3, 1, 7, 2, 1, 5, 2, 3, 1, 4, 2, 3, 8, 1, 7, 2, 3, 2401, 5, 1, 29, 2, 3, 6, 5, 4, 1, 10, 2, 3, 9, 5, 4, 7, 1, 19, 2, 3, 13, 5, 4, 7, 10, 1, 6, 2, 3, 8, 5, 4, 7, 18, 19, 1, 9, 2, 3, 24, 5, 4, 7, 16, 21, 43, 1, 13, 2, 3, 2401, 5, 4, 7, 49, 31213, 9604, 6, 1, 8, 2, 3, 10, 5, 4, 7, 33, 22

Offset: 1

Labos Elemer, Aug 26 2002

In the table output, one can observe constant diagonals (or lines in the square output). The indices of these are: 1, 3, 4, 6, 7, 8, 12, 13, ... (see A002191). And the corresponding values are: 1, 2, 3, 5, 4, 7, 6, 9, ... (see A002192). - Michel Marcus, Dec 19 2013

			Triangle begins
1;
1,3;
1,7,2;
1,5,2,3;
1,4,2,3,8; ...

Cf. A000203, A028982, A074384, A074386, A072461, A072462, A072862.

{k=0, s=0, fl=1}; Table[Print["#"]; Table[fl=1; Print[{r, m}]; Do[s=Mod[DivisorSigma[1, n], m]; If[(s==r)&&(fl==1), Print[n]; fl=0], {n, 1, 150000}], {r, 0, m-1}], {m, 1, 25}]

Min{x; Mod[sigma[x], n]=r}, r=1..n, n=1, ...

A078923 Possible values of sigma(n)-n.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Benoit Cloitre, Dec 15 2002

nonn

To test whether k>1 is in the sequence, it suffices to check values of n up to (k-1)^2, since sigma(n)-n >= sqrt(n)+1 if n is composite.
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
The lower asymptotic density is at least 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 1 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number absent from this sequence.) - Charles R Greathouse IV, Dec 14 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Don Coppersmith, An answer to the problem of Don Saari, 1987.
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.

Cf. A000203, A001065, A002191, A007369. Complement of A005114.

lista(nn)=for (n=0, nn, if (n==1, kmax=2, kmax=(n-1)^2); for (k=1, kmax, if (sigma(k)-k == n, print1(n, ", "); break););); \\ Michel Marcus, Nov 11 2014

Edited by Dean Hickerson, Dec 19 2002

Offset fixed by Michel Marcus, Dec 19 2014

A231368 Possible values for sum of non-divisors of n (A024816).

Original entry on oeis.org

0, 2, 3, 9, 20, 21, 32, 37, 50, 54, 77, 81, 96, 105, 132, 135, 168, 170, 199, 217, 240, 252, 294, 309, 338, 350, 393, 405, 464, 465, 513, 541, 575, 582, 665, 681, 724, 730, 807, 819, 902, 906, 957, 1009, 1052, 1080, 1168, 1182, 1254, 1280, 1365, 1377, 1468

Offset: 1

Jaroslav Krizek, Nov 09 2013

nonn

a(n) = possible values of A024816(m) in increasing order, where A024816(m) = sum of non-divisors of m that are between 1 and m.
Numbers n such that A231367(n) = 1 and A231366(n) >= 1.
Complement of A231369.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A002191 (possible values for sum of divisors of n), A231365, A231366, A231367, A231369, A024816.

A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, May 20 2003

nonn

			29 and 33 are the 15th and 16th nonsense values for sigma(x), since there exist no numbers n of which they are sums of divisors, while {30,31,32} equal sigma(x); e.g., for x = 29, 16, 31, respectively, thus 33 - 29 = 4 = a(15) = A007369(16) - A007369(15).

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

Cf. A002191, A007369, A083235, A083531, A083533, A083534, A083536.

t0[x_] := Table[j, {j, 1, x}]; t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; c=Complement[t0[25000], u]; Delete[c-RotateRight[c], 1]

a(n) = A007369(n+1) - A007369(n).

cp's OEIS Frontend

A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

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A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

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A085790 Integers sorted by the sum of their divisors.

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A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

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A175192 a(n) = characteristic function of numbers k such that A000203(m) = k has solution, where A000203(m) = sums of divisors of m.

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A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

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A074625 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows, where T(n,k) = smallest number x such that Mod[sigma[x],n]=k.

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