A000203 -id:A000203 - cp's OEIS frontend

A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.

2, 4, 8, 8, 12, 16, 16, 16, 26, 24, 24, 32, 28, 32, 48, 32, 36, 52, 40, 48, 64, 48, 48, 64, 62, 56, 80, 64, 60, 96, 64, 64, 96, 72, 96, 104, 76, 80, 112, 96, 84, 128, 88, 96, 156, 96, 96, 128, 114, 124, 144, 112, 108, 160, 144, 128, 160, 120, 120, 192, 124, 128, 208

Offset: 1

Labos Elemer, May 22 2000

Sum of divisors of 2*n that do not divide n. - Franklin T. Adams-Watters, Oct 04 2018

a(n) = 2*n iff n = 2^k, k >= 0 (A000079). - Bernard Schott, Mar 24 2020

			n=9: sigma(18)=18+9+6+3+2+1=39, sigma(9)=9+3+1=13, a(9)=39-13=26.

Paul Tek, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Wang-Sun formula in GL(Z/2kZ), Integers, Vol. 23 (2023), #A37.

Cf. A000079, A000203, A000593, A111003, A002131, A320059.

[DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; Vincenzo Librandi, Oct 05 2018

a:= proc(n) local e;
  e:= 2^padic:-ordp(n,2);
  2*e*numtheory:-sigma(n/e)
end proc:
map(a, [$1..100]); # Robert Israel, Jul 05 2016

Table[DivisorSigma[1,2n]-DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, May 11 2014 *)
Table[CoefficientList[Series[-Log[EllipticTheta[4, 0, x]], {x, 0, 80}],x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)

a(n)=sigma(2*n)-sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

a(n) = A000203(2n) - A000203(n).
a(n) = 2*A002131(n).
a(2*n) = A000203(n) + A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
L.g.f.: -log(EllipticTheta(4,0,x)) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
G.f.: Sum_{k>=1} 2*k*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Oct 23 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). - Amiram Eldar, Jan 19 2024

A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 7, 2, 7, 5, 13, 0, 17, 9, 12, 8, 23, 5, 27, 8, 21, 20, 34, 1, 33, 25, 30, 17, 46, 7, 50, 22, 40, 37, 46, 6, 62, 43, 50, 19, 70, 19, 74, 37, 46, 55, 82, 9, 79, 46, 70, 47, 95, 32, 83, 38, 81, 74, 107, 2, 112, 81, 76, 56, 102, 45, 125, 70, 103, 58, 133, 14, 138, 101

Offset: 1

N. J. A. Sloane, Oct 12 2000

Theorem (Lagarias): a(n) is nonnegative for all n if and only if the Riemann Hypothesis is true.
Up to rank n=10^4, zeros occur only at n=1,2,4,6 and 12; ones occur at n=3 and n=24. The first occurrence of k = 0,1,2,3,... is at n = 1,3,8,-1,5,10,36,7,16,14,-1,-1,15,11,72,... where -1 means that k does not occur among the first 10^4 terms. - Robert G. Wilson v, Dec 06 2010, reformulated by M. F. Hasler, Sep 09 2011
Looking at the graph of this sequence, it appears that there is a slowly growing lower bound. It is even more apparent when larger ranges of points are computed. Numbers A176679(n+2) and A222761(n) give the (x,y) coordinates of the n-th point. - T. D. Noe, Mar 28 2013

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Peter Luschny, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.

Cf. A057640, A000203, A076633, A067698, A079526, A058209.

f[n_] := Block[{h = HarmonicNumber@n}, Floor[h + Exp@h*Log@h] - DivisorSigma[1, n]]; Array[f, 74] (* Robert G. Wilson v, Dec 06 2010 *)

a(n)={my(H=sum(k=1,n,1/k)); floor(exp(H)*log(H)+H) - sigma(n)}
list_A057641(Nmax,H=0,S=1)=for(n=S,Nmax, H+=1/n; print1(floor(exp(H)*log(H)+H) - sigma(n),","))  \\ M. F. Hasler, Sep 09 2011

a(n) = A057640(n) - A000203(n). - Omar E. Pol, Oct 25 2019

Five more terms from Robert G. Wilson v, Dec 06 2010

I deleted some unproved assertions by Robert G. Wilson v about the presence of 0's, 1's, ... in this sequence. - N. J. A. Sloane, Dec 07 2010

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

Original entry on oeis.org

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

A286603 Restricted growth sequence computed for sigma, A000203.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 14, 17, 21, 22, 23, 24, 18, 25, 26, 27, 26, 28, 29, 20, 22, 30, 17, 31, 32, 33, 34, 24, 26, 35, 36, 37, 24, 38, 27, 39, 24, 39, 40, 30, 20, 41, 42, 31, 43, 44, 33, 45, 46, 47, 31, 45, 24, 48, 49, 50, 35, 51, 31, 41, 40, 52, 53, 47, 33, 54, 55, 56, 39, 57, 30, 58, 59, 41, 60

Offset: 1

Antti Karttunen, May 11 2017

nonn

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A000203, because for all i, j it holds that: a(i) = a(j) <=> A000203(i) = A000203(j) <=> A286358(i) = A286358(j).

Note that the latter equivalence indicates that this is also the restricted growth sequence of A286358.

			Construction: we start with a(1)=1 for sigma(1)=1 (where sigma = A000203), and then after, for all n > 1, whenever the value of sigma(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever sigma(n) = sigma(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, sigma(2) = 3, not encountered before, thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n=3, sigma(3) = 4, not encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=4, sigma(4) = 7, not encountered before, thus we allot for a(4) the least so far unused number, which is 4, thus a(4) = 4.
For n=5, sigma(5) = 6, not encountered before, thus we allot for a(5) the least so far unused number, which is 5, thus a(5) = 5.
For n=6, sigma(6) = 12, not encountered before, thus we allot for a(6) the least so far unused number, which is 6, thus a(6) = 6.
And this continues for n=7..10 because also for those n sigma obtains fresh new values, so here a(n) = n up to n = 10.
But then comes n=11, where sigma(11) = 12, a value which was already encountered at n=6 for the first time, thus we set a(11) = a(6) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A206036, A211656, A286358.

Cf. also A101296, A286605, A286610, A286619, A286621, A286622, A286626, A286378.

With[{nn = 93}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

A000203(n) = sigma(n);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(10000,n,A000203(n))),"b286603.txt");

A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 1, 7, 3, 1, 6, 4, 3, 1, 1, 12, 7, 4, 3, 3, 1, 1, 8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1, 15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 18, 15, 8, 12, 12, 6, 6, 7, 7, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2020

The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - Omar E. Pol, Jan 20 2022

			Triangle begins:
   1;
   3;
   4,  1;
   7,  3,  1;
   6,  4,  3, 1, 1;
  12,  7,  4, 3, 3, 1, 1;
   8,  6,  7, 4, 4, 3, 3, 1, 1, 1, 1;
  15, 12,  6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
  13,  8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From _Omar E. Pol_, Jan 13 2022: (Start)
Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
    Level k: 1              2         3        4       5      6     7
Stage
  n   _ _ _ _ _ _ _ _
     |            _  |
  1  |           |_| |
     |_ _ _ _ _ _ _ _|
     |          _    |
     |         | |_  |
  2  |         |_ _| |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _
     |        _      |        _  |
     |       | |     |       |_| |
  3  |       |_|_ _  |           |
     |         |_ _| |           |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _
     |      _        |      _    |      _  |
     |     | |       |     | |_  |     |_| |
  4  |     | |_      |     |_ _| |         |
     |     |_  |_ _  |           |         |
     |       |_ _ _| |           |         |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
     |    _          |    _      |    _    |    _  |    _  |
     |   | |         |   | |     |   | |_  |   |_| |   |_| |
     |   | |         |   |_|_ _  |   |_ _| |       |       |
  5  |   |_|_        |     |_ _| |         |       |       |
     |       |_ _ _  |           |         |       |       |
     |       |_ _ _| |           |         |       |       |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
     |  _            |  _        |  _      |  _    |  _    |  _  |  _  |
     | | |           | | |       | | |     | | |_  | | |_  | |_| | |_| |
     | | |           | | |_      | |_|_ _  | |_ _| | |_ _| |     |     |
     | | |_ _        | |_  |_ _  |   |_ _| |       |       |     |     |
  6  | |_    |       |   |_ _ _| |         |       |       |     |     |
     |   |_  |_ _ _  |           |         |       |       |     |     |
     |     |_ _ _ _| |           |         |       |       |     |     |
     |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
.
Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
For n = 6 and k = 3 we add four cubes at 6th stage in the third level of the structure of the tower starting from the base so T(6,3) = 4.
For n = 9 another connection with the tower is as follows:
First we take the columns from the above triangle and build a new triangle in which all columns start at row 1 as shown below:
.
   1,  1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
   3,  3,  3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
   4,  4,  4, 4, 4, 4, 4, 4, 4, 4, 4;
   7,  7,  7, 7, 7, 7, 7;
   6,  6,  6, 6, 6;
  12, 12, 12;
   8,  8;
  15;
  13;
.
Then we rotate the triangle by 90 degrees as shown below:
                                       _
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  | |
  1;                                  |_|_
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |   |
  1, 3;                               |_ _|_
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |   | |
  1, 3, 4;                            |_ _|_|_
  1, 3, 4, 7;                         |     | |
  1, 3, 4, 7;                         |_ _ _| |_
  1, 3, 4, 7, 6;                      |     |   |
  1, 3, 4, 7, 6;                      |_ _ _|_ _|_
  1, 3, 4, 7, 6, 12;                  |_ _ _ _| | |_
  1, 3, 4, 7, 6, 12, 8;               |_ _ _ _|_|_ _|_ _
  1, 3, 4, 7, 6, 12, 8, 15; 13;       |_ _ _ _ _|_ _|_ _|
.
                                         Lateral view
                                         of the tower
.                                      _ _ _ _ _ _ _ _ _
                                      |_| | | | | | |   |
                                      |_ _|_| | | | |   |
                                      |_ _|  _|_| | |   |
                                      |_ _ _|    _|_|   |
                                      |_ _ _|  _|    _ _|
                                      |_ _ _ _|     |
                                      |_ _ _ _|  _ _|
                                      |         |
                                      |_ _ _ _ _|
.
                                           Top view
                                         of the tower
.
The sum of the m-th row of the new triangle equals A024916(j) where j is the length of the m-th row, equaling the number of cubic cells in the m-th level of the tower. For example: the last row of triangle has 9 terms and the sum of the last row is 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 13 = A024916(9) = 69, equaling the number of cubes in the base of the tower. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened)

Sum of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000203.
The length of the m-th block in row n is A187219(m), m >= 1.
Row sums give A138879.
Cf. A337209 (another version).
Cf. A272172 (analog for the stepped pyramid described in A245092).
Cf. A024916, A135010, A138121, A138137, A138785, A221529, A235791, A236104, A237270, A237591, A237593, A238442, A338156, A340423, A345023.

A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
A339278[15] (* Generates 15 rows *) (* Paolo Xausa, Feb 17 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

A339278(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,sigma(n-m+1)))));
A339278(15) \\ Generates 15 rows \\ Paolo Xausa, Feb 17 2023

a(m) = A000203(A336811(m)).

T(n,k) = A000203(A336811(n,k)).

A062700 Terms of A000203 that are prime.

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jul 11 2001

nonn

Sorted and duplicates removed, this gives A023195.

			sigma(2) = 3, sigma(4) = 7, sigma(9) = 13 are the first three prime terms of A000203. Hence the sequence starts 3, 7, 13.

Harry J. Smith and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harry J. Smith)

Cf. A000203 (sigma(n), sum of divisors of n), A023194, A034885 (record values of sigma(n)), A023195 (prime numbers that are the sum of the divisors of some n), A100382 (record values of A062700).

[ c: n in [1..1000000] | IsPrime(c) where c:=SumOfDivisors(n) ]; // Klaus Brockhaus, Oct 21 2009

Select[DivisorSigma[1,Range[1000000]],PrimeQ] (* Harvey P. Dale, Nov 09 2012 *)

je=[]; for(n=1,1000000, if(isprime(sigma(n)),je=concat(je, sigma(n)))); je

{ n=0; for (m=1, 10^9, if(isprime(a=sigma(m)), write("b062700.txt", n++, " ", a); if (n==100, break)) ) } \\ Harry J. Smith, Aug 09 2009

from sympy import isprime, divisor_sigma
A062700_list = [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

a(n) = A000203(A023194(n)). - Michel Marcus, Oct 19 2019

Edited by Klaus Brockhaus, Oct 21 2009

A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

Offset: 1

Antti Karttunen and Peter Munn, May 26 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.
A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.
If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.
Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).
Cf. also A048250, A291750, A291751, A340070, A323363.
Subsets of range: A008864, A065091 (conjectured).

Table[GCD[DivisorSigma[1,n],DivisorSum[n,MoebiusMu[n/#]^2*#&]],{n,100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
(Python 3.8+)
from math import prod, gcd
from sympy import primefactors, divisor_sigma
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
def A344695(n): return gcd(A001615(n),divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021

a(n) = gcd(A000203(n), A001615(n)).
For prime p, a(p^e) = (p+1)^(e mod 2).
For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
a(A007913(n)) | a(n).
a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).
a(n) = A000203(n) / A344696(n).
a(n) = A001615(n) / A344697(n).

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

A054799 Integers n such that sigma(n+2) = sigma(n) + 2, where sigma = A000203, the sum of divisors of n.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 434, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487

Offset: 1

Labos Elemer, May 22 2000

nonn

Only 3 composite numbers are known: 434, 8575, 8825. This sequence is the union of A050507 and A001359.

The terms are also the solutions of A001065(x) = A001065(x+2), where A001065(n) is the sum of proper divisors of n. - Michel Marcus, Nov 14 2014

			n = 434, divisors = {1, 2, 7, 14, 31, 62, 217, 434}, sigma(434) = 768, sigma(436) = 770; n = 8575, divisors = {1, 5, 7, 25, 35, 49, 175, 245, 343, 1225, 1715, 8575}, sigma(8575) = 12400, sigma(8577) = 12402; n = 8825, divisors = {1, 5, 25, 353, 1765, 8825}, sigma(8525) = 10974, sigma(8527) = 10976.

Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions., M.Dekker Inc., New York, Problem 12 in Chapter V., p. 81.

Cf. A000203, A001359, A050507.

Select[Range[1500],DivisorSigma[1,#+2]==DivisorSigma[1,#]+2&] (* Jayanta Basu, May 01 2013 *)

is(n)=sigma(n+2)==sigma(n)+2 \\ Charles R Greathouse IV, Feb 13 2013

A056550 Numbers k such that Sum_{j=1..k} sigma(j) is divisible by k, where sigma(j) = sum of divisors of j (A000203).

Original entry on oeis.org

1, 2, 8, 11, 17, 63, 180, 259, 818, 2161, 4441, 8305, 11998, 694218, 3447076, 4393603, 57402883, 73459800, 121475393, 2068420025, 5577330586, 13320495021, 35297649260, 138630178659, 988671518737, 1424539472772, 3028785109162, 13702718147734, 21320824383487

Offset: 1

Asher Auel, Jun 06 2000

			a(3) = 8 is in the sequence because A024916(8) / 8 = 56 / 8 = 7 is an integer. [_Jaroslav Krizek_, Dec 07 2009]

Cf. A000203, A024916, A168133, A168132.

f := []: for i from 1 to 9000 do if add(sigma(n), n=1..i) mod i = 0 then f := [op(f),i] fi; od; f;

k=10^4;a[1]=1;a[n_]:=a[n]=DivisorSigma[1,n]+a[n-1]; s=a/@Range@k;Select[Range@k,Divisible[s[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=44*10^5,ds},ds=Accumulate[DivisorSigma[1,Range[nn]]];Select[ Thread[{ds,Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[All,2]] (* The program generates the first 16 terms of the sequence. To generate more, increase the value of nn. *) (* Harvey P. Dale, Dec 04 2018 *)

is(n)=sum(k=1,n,n\k*k)%n==0 \\ Charles R Greathouse IV, Feb 14 2013

Values of k for which A024916(k)/k is an integer.

More terms from Jud McCranie, Jul 04 2000
a(19)-a(24) from Donovan Johnson, Dec 29 2008
a(25) from Donovan Johnson, Jun 16 2011
a(26) from Jud McCranie, Dec 17 2024
a(27) from Jud McCranie, Dec 22 2024
a(28) from Jud McCranie, Apr 03 2025
a(29) from Jud McCranie, May 04 2025

cp's OEIS Frontend

A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.

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A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

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

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A286603 Restricted growth sequence computed for sigma, A000203.

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A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.

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A062700 Terms of A000203 that are prime.

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