A326124 -id:A326124 - cp's OEIS frontend

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

1, 4, 8, 15, 21, 33, 41, 56, 69, 87, 99, 127, 141, 165, 189, 220, 238, 277, 297, 339, 371, 407, 431, 491, 522, 564, 604, 660, 690, 762, 794, 857, 905, 959, 1007, 1098, 1136, 1196, 1252, 1342, 1384, 1480, 1524, 1608, 1686, 1758, 1806, 1930, 1987, 2080, 2152

Offset: 1

Clark Kimberling

Row sums of triangle A128489. E.g., a(5) = 15 = (10 + 3 + 1 + 1), sum of row 4 terms of triangle A128489. - Gary W. Adamson, Jun 03 2007
Row sums of triangle A134867. - Gary W. Adamson, Nov 14 2007
a(10^4) = 82256014, a(10^5) = 8224740835, a(10^6) = 822468118437, a(10^7) = 82246711794796; see A072692. - M. F. Hasler, Nov 22 2007
Equals row sums of triangle A158905. - Gary W. Adamson, Mar 29 2009
n is prime if and only if a(n) - a(n-1) - 1 = n. - Omar E. Pol, Dec 31 2012
Also the alternating row sums of A236104. - Omar E. Pol, Jul 21 2014
a(n) is also the total number of parts in all partitions of the positive integers <= n into equal parts. - Omar E. Pol, Apr 30 2017
a(n) is also the total area of the terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Nov 04 2017
a(n) is also the area under the Dyck path described in the n-th row of A237593 (see example). - Omar E. Pol, Sep 17 2018
From Omar E. Pol, Feb 17 2020: (Start)
Convolution of A340793 and A000027.
Convolved with A340793 gives A000385. (End)
a(n) is also the number of cubic cells (or cubes) in the n-th level starting from the top of the stepped pyramid described in A245092. - Omar E. Pol, Jan 12 2022

			From _Omar E. Pol_, Aug 20 2021: (Start)
For n = 6 the sum of all divisors of the first six positive integers is [1] + [1 + 2] + [1 + 3] + [1 + 2 + 4] + [1 + 5] + [1 + 2 + 3 + 6] = 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33.
On the other hand the area under the Dyck path of the 6th diagram as shown below is equal to 33, so a(6) = 33.
Illustration of initial terms:                        _ _ _ _
                                        _ _ _        |       |_
                            _ _ _      |     |       |         |_
                  _ _      |     |_    |     |_ _    |           |
          _ _    |   |_    |       |   |         |   |           |
    _    |   |   |     |   |       |   |         |   |           |
   |_|   |_ _|   |_ _ _|   |_ _ _ _|   |_ _ _ _ _|   |_ _ _ _ _ _|
.
    1      4        8          15           21             33         (End)

Hardy and Wright, "An introduction to the theory of numbers", Oxford University Press, fifth edition, p. 266.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Plot of (a(n) - Pi^2*n^2/12) / (n*log(n)^(2/3)) for n = 2..100000.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916.
Peter Polm, C# program for A024916.
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.

Partial sums of A000203.
Cf. A056550, A104471(2*n-1, n), A123229, A128489, A000217, A134867, A072692, A158905, A237593, A245092, A006218, A222548, A092406, A160664.
Cf. A000385, A010766, A340793.

a024916 n = sum $ map (\k -> k * div n k) [1..n]
-- Reinhard Zumkeller, Apr 20 2015

[(&+[DivisorSigma(1, k): k in [1..n]]): n in [1..60]]; // G. C. Greubel, Mar 15 2019

A024916 := proc(n)
    add(numtheory[sigma](k),k=0..n) ;
end proc: # Zerinvary Lajos, Jan 11 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      numtheory[sigma](n)+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 12 2019

Table[Plus @@ Flatten[Divisors[Range[n]]], {n, 50}] (* Alonso del Arte, Mar 06 2006 *)
Table[Sum[n - Mod[n, m], {m, n}], {n, 50}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2006 *)
a[n_] := Sum[DivisorSigma[1, k], {k, n}]; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 16 2011 *)
Accumulate[DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Mar 13 2014 *)

A024916(n)=sum(k=1,n,n\k*k) \\ M. F. Hasler, Nov 22 2007

A024916(z) = { my(s,u,d,n,a,p); s = z*z; u = sqrtint(z); p = 2; for(d=1, u, n = z\d - z\(d+1); if(n<=1, p=d; break(), a = z%d; s -= (2*a+(n-1)*d)*n/2); ); u = z\p; for(d=2, u, s -= z%d); return(s); } \\ See the link for a nicely formatted version. - P. L. Patodia (pannalal(AT)usa.net), Jan 11 2008

A024916(n)={my(s=0,d=1,q=n);while(dPeter Polm, Aug 18 2014

A024916(n)={ my(s=n^2, r=sqrtint(n), nd=n, D); for(d=1, r, (1>=D=nd-nd=n\(d+1)) && (r=d-1) && break; s -= n%d*D+(D-1)*D\2*d); s - sum(d=2, n\(r+1), n%d)} \\ Slightly optimized version of Patodia's code. - M. F. Hasler, Apr 18 2015
(C#) See Polm link.

def A024916(n): return sum(k*(n//k) for k in range(1,n+1)) # Chai Wah Wu, Dec 17 2021

from math import isqrt
def A024916(n): return (-(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 21 2023

[sum(sigma(k) for k in (1..n)) for n in (1..60)] # G. C. Greubel, Mar 15 2019

From Benoit Cloitre, Apr 28 2002: (Start)
a(n) = n^2 - A004125(n).
Asymptotically a(n) = n^2*Pi^2/12 + O(n*log(n)). (End)
G.f.: (1/(1-x))*Sum_{k>=1} x^k/(1-x^k)^2. - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{m=1..n} (n - (n mod m)). - Roger L. Bagula and Gary W. Adamson, Oct 06 2006
a(n) = n^2*Pi^2/12 + O(n*log(n)^(2/3)) [Walfisz]. - Charles R Greathouse IV, Jun 19 2012
a(n) = A000217(n) + A153485(n). - Omar E. Pol, Jan 28 2014
a(n) = A000292(n) - A076664(n), n > 0. - Omar E. Pol, Feb 11 2014
a(n) = A078471(n) + A271342(n). - Omar E. Pol, Apr 08 2016
a(n) = (1/2)*(A222548(n) + A006218(n)). - Ridouane Oudra, Aug 03 2019
From Greg Dresden, Feb 23 2020: (Start)
a(n) = A092406(n) + 8, n>3.
a(n) = A160664(n) - 1, n>0. (End)
a(2*n) = A326123(n) + A326124(n). - Vaclav Kotesovec, Aug 18 2021
a(n) = Sum_{k=1..n} k * A010766(n,k). - Georg Fischer, Mar 04 2022

A062731 Sum of divisors of 2*n.

Original entry on oeis.org

3, 7, 12, 15, 18, 28, 24, 31, 39, 42, 36, 60, 42, 56, 72, 63, 54, 91, 60, 90, 96, 84, 72, 124, 93, 98, 120, 120, 90, 168, 96, 127, 144, 126, 144, 195, 114, 140, 168, 186, 126, 224, 132, 180, 234, 168, 144, 252, 171, 217, 216, 210, 162, 280, 216, 248, 240, 210

Offset: 1

Jason Earls, Jul 11 2001

a(n) is also the total number of parts in all partitions of 2*n into equal parts. - Omar E. Pol, Feb 14 2021

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Harry J. Smith]

Sigma(k*n): A000203 (k=1), A144613 (k=3), A193553 (k=4, even bisection), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A008438, A074400, A182818, A239052 (odd bisection), A326124 (partial sums), A054784, A215947, A336923, A346870, A346878, A346880, A355750.
Row 2 of A319526. Column & Row 2 of A216626. Row 1 of A355927.
Shallow diagonal (2n,n) of A265652. See also A244658.

[SumOfDivisors(2*n): n in [1..70]]; // Vincenzo Librandi, Oct 31 2014

lst={};Do[AppendTo[lst, DivisorSigma[1, n]], {n, 2, 6!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)
DivisorSigma[1,2*Range[60]] (* Harvey P. Dale, Jun 08 2022 *)

numlib::sigma(2*n)$ n=0..81 // Zerinvary Lajos, May 13 2008

PARI
```
vector(66,n,sigma(2*n,1))
```

for (n=1, 1000, write("b062731.txt", n, " ", sigma(2*n)) ) \\ Harry J. Smith, Aug 09 2009

a(n) = A000203(2*n). - R. J. Mathar, Apr 06 2011
a(n) = A000203(n) + A054785(n). - R. J. Mathar, May 19 2020
From Vaclav Kotesovec, Aug 07 2022: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * (3 - 2^(1-s)).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. (End)
From Miles Wilson, Sep 30 2024: (Start)
G.f.: Sum_{k>=1} k*x^(k/gcd(k, 2))/(1 - x^(k/gcd(k, 2))).
G.f.: Sum_{k>=1} k*x^(2*k/(3 + (-1)^k))/(1 - x^(2*k/(3 + (-1)^k))). (End)

Zero removed and offset corrected by Omar E. Pol, Jul 17 2009

A078471 Sum of all odd divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 6, 7, 13, 17, 25, 26, 39, 45, 57, 61, 75, 83, 107, 108, 126, 139, 159, 165, 197, 209, 233, 237, 268, 282, 322, 330, 360, 384, 416, 417, 465, 483, 531, 544, 582, 602, 658, 664, 706, 738, 782, 794, 872, 896, 944, 948, 1005, 1036, 1108, 1122, 1176, 1216

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

The subsequence of primes begins: 2, 7, 13, 17, 61, 83, 107, 139, 197, 233, then no more through a(54). [Jonathan Vos Post, Feb 14 2010]

a(n) is also the total number of parts in all partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Partial sums of A000593.

Cf. A024916, A271342, A000203, A326124.

[&+[&+[d:d in Divisors(k)|IsOdd(d)]:k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Aug 28 2019

with(numtheory):
b:= n-> add(d, d=select(x-> x::odd, divisors(n))):
a:= proc(n) option remember; b(n)+`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 25 2015

a[n_] := Sum[DivisorSum[k, (-1)^(# + 1) k/# &], {k, 1, n}]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)
Accumulate[Table[Total[Select[Divisors[n],OddQ]],{n,60}]] (* Harvey P. Dale, Sep 15 2024 *)

a(n)=sum(v=1,n,sumdiv(v,d,(-1)^(d+1)*v/d))

a(n) = sum(k=1, n, sumdiv(k, d, (d%2)*d)); \\ Michel Marcus, Apr 09 2016

def A078471(n): return sum(k*(n//k) for k in range((n>>1)+1, n+1)) + sum(k*(n//k-((n>>1)//k<<1)) for k in range(1, (n>>1)+1)) # Chai Wah Wu, Apr 26 2023

from math import isqrt
def A078471(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} A000593(k).
a(n) is asymptotic to c*n^2 where c = Pi^2/24.
a(n) = A024916(n) - A271342(n). - Omar E. Pol, Apr 08 2016
G.f.: (1/(1 - x))*Sum_{k>=1} k*x^k/(1 + x^k). - Ilya Gutkovskiy, Dec 23 2016
From Ridouane Oudra, Aug 28 2019: (Start)
a(n) = Sum_{k=1..n} (sigma(2k) - 2*sigma(k)), where sigma = A000203.
a(n) = A326124(n) - 2*A024916(n). (End)

Better definition from Omar E. Pol, Apr 09 2016

A326123 a(n) is the sum of all divisors of the first n odd numbers.

Original entry on oeis.org

1, 5, 11, 19, 32, 44, 58, 82, 100, 120, 152, 176, 207, 247, 277, 309, 357, 405, 443, 499, 541, 585, 663, 711, 768, 840, 894, 966, 1046, 1106, 1168, 1272, 1356, 1424, 1520, 1592, 1666, 1790, 1886, 1966, 2087, 2171, 2279, 2399, 2489, 2601, 2729, 2849, 2947, 3103, 3205, 3309, 3501, 3609, 3719

Offset: 1

Omar E. Pol, Jun 07 2019

a(n)/A326124(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n odd-indexed levels of the stepped pyramid described in A245092.

			For n = 3 the first three odd numbers are [1, 3, 5] and their divisors are [1], [1, 3], [1, 5] respectively, and the sum of these divisors is 1 + 1 + 3 + 1 + 5 = 11, so a(3) = 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Partial sums of A008438.

Cf. A000203, A005408, A024916, A237593, A245092, A326124.

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=1..200,2)])); # Robert Israel, Jun 12 2019

Accumulate@ DivisorSigma[1, Range[1, 109, 2]] (* Michael De Vlieger, Jun 09 2019 *)

terms(n) = my(s=0, i=0); for(k=0, n-1, if(i>=n, break); s+=sigma(2*k+1); print1(s, ", "); i++)
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Jun 08 2019

a(n) = sum(k=1, 2*n-1, if (k%2, sigma(k))); \\ Michel Marcus, Jun 08 2019

from math import isqrt
def A326123(n): return (-(s:=isqrt(r:=n<<1))**2*(s+1) + sum((q:=r//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) -(t:=isqrt(m:=n>>1))**2*(t+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))+3*((u:=isqrt(n))**2*(u+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,u+1))>>1) # Chai Wah Wu, Nov 01 2023

a(n) = A024916(2n) - A326124(n).

a(n) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 18 2021

A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

Original entry on oeis.org

1, 3, 6, 7, 8, 16, 10, 15, 21, 22, 14, 36, 16, 28, 42, 31, 20, 55, 22, 50, 54, 40, 26, 76, 43, 46, 66, 64, 32, 108, 34, 63, 78, 58, 74, 123, 40, 64, 90, 106, 44, 140, 46, 92, 144, 76, 50, 156, 73, 117, 114, 106, 56, 172, 106, 136, 126, 94, 62, 240, 64, 100, 186, 127

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of aliquot divisors (or aliquot parts) of the n-th positive even number.

a(n) has a symmetric representation.

			For n = 5 the 5th even number is 10 and the divisors of 10 are [1, 2, 5, 10] and the sum of the divisors of 10 except for the largest is 1 + 2 + 5 = 8, so a(5) = 8.

Bisection of A001065.
Partial sums give A347154.
Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

a[n_] := DivisorSigma[1, 2*n] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 20 2021 *)

a(n) = sigma(2*n) - 2*n; \\ Michel Marcus, Aug 20 2021

from sympy import divisors
def a(n): return sum(divisors(2*n)[:-1])
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Aug 20 2021

a(n) = A001065(2*n).
a(n) = 1 + A346880(n).
Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 17 2024

A346870 Sum of all divisors, except the smallest and the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

0, 2, 7, 13, 20, 35, 44, 58, 78, 99, 112, 147, 162, 189, 230, 260, 279, 333, 354, 403, 456, 495, 520, 595, 637, 682, 747, 810, 841, 948, 981, 1043, 1120, 1177, 1250, 1372, 1411, 1474, 1563, 1668, 1711, 1850, 1895, 1986, 2129, 2204, 2253, 2408, 2480, 2596, 2709, 2814

Offset: 1

Omar E. Pol, Aug 18 2021

Partial sums of the even-indexed terms of Chowla's function A048050.

a(n) has a symmetric representation.

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

Partial sums of A346880.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346869.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+numtheory[sigma](2*n)-1-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021

s[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
from itertools import accumulate
def A346880(n): return sum(divisors(2*n)[1:-1])
def aupton(nn): return list(accumulate(A346880(n) for n in range(1, nn+1)))
print(aupton(52)) # Michael S. Branicky, Aug 19 2021

from math import isqrt
def A346870(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+2) # Chai Wah Wu, Nov 02 2023

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

A346880 Sum of the divisors, except the smallest and the largest, of the n-th positive even number.

Original entry on oeis.org

0, 2, 5, 6, 7, 15, 9, 14, 20, 21, 13, 35, 15, 27, 41, 30, 19, 54, 21, 49, 53, 39, 25, 75, 42, 45, 65, 63, 31, 107, 33, 62, 77, 57, 73, 122, 39, 63, 89, 105, 43, 139, 45, 91, 143, 75, 49, 155, 72, 116, 113, 105, 55, 171, 105, 135, 125, 93, 61, 239, 63, 99, 185, 126, 121

Offset: 1

Omar E. Pol, Aug 18 2021

a(n) has a symmetric representation.

			For n = 5 the 5th even number is 10 and the divisors of 10 are [1, 2, 5, 10] and the sum of the divisors of 10 except the smaller and the largest is 2 + 5 = 7, so a(5) = 7.

Bisection of A048050.
Partial sums give A346870.
Cf. A000203, A005843, A062731, A237593, A245092, A244049, A326124, A346879.

a[n_] := DivisorSigma[1, 2*n] - 2*n - 1; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
def a(n): return sum(divisors(2*n)[1:-1])
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Aug 19 2021

a(n) = A048050(2*n).

Sum_{k=1..n} a(k) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024

A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

1, 4, 10, 17, 25, 41, 51, 66, 87, 109, 123, 159, 175, 203, 245, 276, 296, 351, 373, 423, 477, 517, 543, 619, 662, 708, 774, 838, 870, 978, 1012, 1075, 1153, 1211, 1285, 1408, 1448, 1512, 1602, 1708, 1752, 1892, 1938, 2030, 2174, 2250, 2300, 2456, 2529, 2646, 2760

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of all aliquot divisors (or aliquot parts) of the first n positive even numbers.
Partial sums of the even-indexed terms of A001065.
a(n) has a symmetric representation.

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

Partial sums of A346878.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

s[n_] := DivisorSigma[1, 2*n] - 2*n; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)

a(n) = sum(k=1, n, k*=2; sigma(k)-k); \\ Michel Marcus, Aug 20 2021

from sympy import divisors
from itertools import accumulate
def A346878(n): return sum(divisors(2*n)[:-1])
def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))
print(aupton(51)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
def A347154(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+1) # Chai Wah Wu, Nov 02 2023

a(n) = n + A346870(n).

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

cp's OEIS Frontend

A024916 a(n) = Sum_{k=1..n} k*floor(n/k); also Sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n (A000203).

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A062731 Sum of divisors of 2*n.

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A078471 Sum of all odd divisors of all positive integers <= n.

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A326123 a(n) is the sum of all divisors of the first n odd numbers.

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A346878 Sum of the divisors, except for the largest, of the n-th positive even number.

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A347108 a(n) = Sum_{k=1..n} sigma(k)sigma(2k), where sigma(n) = A000203(n) is the sum of the divisors of n.