A143128 -id:A143128 - cp's OEIS frontend

A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

1, 5, 13, 28, 49, 82, 123, 179, 248, 335, 434, 561, 702, 867, 1056, 1276, 1514, 1791, 2088, 2427, 2798, 3205, 3636, 4127, 4649, 5213, 5817, 6477, 7167, 7929, 8723, 9580, 10485, 11444, 12451, 13549, 14685, 15881, 17133, 18475, 19859, 21339, 22863, 24471, 26157

Offset: 1

Jaroslav Krizek, Mar 14 2010

Partial sums of A024916. - Omar E. Pol, Jul 03 2014
a(n) is also the volume of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Aug 12 2015
Also the alternating row sums of A262612. - Omar E. Pol, Nov 23 2015
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A000027.
Convolution of A340793 and the nonzero terms of A000217.
Antidiagonal sums of A319073.
Row sums of A274824. (End)
Row sums of A345272. - Omar E. Pol, Jun 14 2021
Also the alternating row sums of A353690. - Omar E. Pol, Jun 05 2022

			For n = 4: a(4) = sigma(1)*4 + sigma(2)*3 + sigma(3)*2 + sigma(4)*1 = 1*4 + 3*3 + 4*2 + 7*1 = 28.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2209 terms from Indranil Ghosh)

Cf. A000203, A000217, A006218, A024916, A072481, A237593, A244050, A245092, A262612, A274824, A319073, A340793, A345272, A353690.

Cf. A143128, A353908.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[sigma](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 07 2021

Table[Sum[DivisorSigma[1, k] (n - k + 1), {k, n}], {n, 45}] (* Michael De Vlieger, Nov 24 2015 *)

a(n) = sum(x=1, n, sigma(x)*(n-x+1)) \\ Michel Marcus, Mar 18 2013

from math import isqrt
def A175254(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*(2*s+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+2*q+1)+3*(n+1)*(2*k+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Conjecture: a(n) = Sum_{k=0..n} A006218(n-k). - R. J. Mathar, Oct 17 2012
a(n) = A000330(n) - A072481(n). - Omar E. Pol, Aug 12 2015
a(n) ~ Pi^2*n^3/36. - Vaclav Kotesovec, Sep 25 2016
G.f.: (1/(1 - x)^2)*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) = Sum_{k=1..n} Sum_{i=1..k} k - (k mod i). - Wesley Ivan Hurt, Sep 13 2017
a(n) = A244050(n)/4. - Omar E. Pol, Jan 22 2021
a(n) = (n+1)*A024916(n) - A143128(n). - Vaclav Kotesovec, May 11 2022

Corrected by Jaroslav Krizek, Mar 17 2010

More terms from Michel Marcus, Mar 18 2013

A244580 Square array read by antidiagonals related to the symmetric representation of sigma.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 4, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 6, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 6, 6, 6, 7, 8, 9, 10, 9, 8, 7, 8, 8, 7, 8, 9, 10, 11, 10, 9, 8, 8, 8, 8, 8, 9, 10, 11, 12, 11, 10, 9, 8, 9, 9, 8, 9, 10, 11, 12

Offset: 1

Omar E. Pol, Jul 04 2014

The number of parts k in the square array is equal to A000203(k) hence the sum of parts k is equal to A064987(k).
The structure has a three-dimensional representation using polycubes. T(n,k) is the height of a column. The total area in the horizontal level z gives A000203(z).
The main diagonal gives A244367.

			.                         _ _ _ _ _ _ _ _ _
1,2,3,4,5,6,7,8,9...     |_| | | | | | | | |
2,2,3,4,5,6,7,8,9...     |_ _|_| | | | | | |
3,3,4,4,5,6,7,8,9...     |_ _|  _|_| | | | |
4,4,4,6,6,6,7,8,9...     |_ _ _|    _|_| | |
5,5,5,6,6,8,8,8,9...     |_ _ _|  _|  _ _|_|
6,6,6,6,8,8,9...         |_ _ _ _|  _| |
7,7,7,7,8,9,9...         |_ _ _ _| |_ _|
8,8,8,8,8...             |_ _ _ _ _|
9,9,9,9,9...             |_ _ _ _ _|
.

Cf. A000203, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A244367.

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A000441 a(n) = Sum_{k=1..n-1} ksigma(k)sigma(n-k).

Original entry on oeis.org

0, 1, 9, 34, 95, 210, 406, 740, 1161, 1920, 2695, 4116, 5369, 7868, 9690, 13640, 16116, 22419, 25365, 34160, 38640, 50622, 55154, 73320, 77225, 100100, 107730, 135576, 141085, 182340, 184760, 233616, 243408, 297738, 301420, 385110, 377511, 467210, 478842

Offset: 1

N. J. A. Sloane

nonn

Apart from initial zero this is the convolution of A340793 and A143128. - Omar E. Pol, Feb 16 2021

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).
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000385, A000477, A000499, A259692, A259693, A259694, A259695, A259696.
Cf. A000203 (sigma_1), A001158 (sigma_3), A064987.
Cf. A143128, A340793.

S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1);
f:=e->[seq(S(n,e),n=1..30)];f(1); # N. J. A. Sloane, Jul 03 2015

a[n_] := Sum[k*DivisorSigma[1, k]*DivisorSigma[1, n-k], {k, 1, n-1}]; Array[a, 40] (* Jean-François Alcover, Feb 08 2016 *)

a(n) = sum(k=1, n-1, k*sigma(k)*sigma(n-k)); \\ Michel Marcus, Feb 02 2014

a(n) = my(f = factor(n)); ((n - 6*n^2) * sigma(f) + 5*n * sigma(f, 3)) / 24; \\ Amiram Eldar, Jan 04 2025

from sympy import divisor_sigma
def A000441(n): return (n*(1-6*n)*divisor_sigma(n)+5*n*divisor_sigma(n,3))//24 # Chai Wah Wu, Jul 25 2024

Convolution of A000203 with A064987. - Sean A. Irvine, Nov 14 2010
G.f.: x*f(x)*f'(x), where f(x) = Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 28 2018
a(n) = (n/24 - n^2/4)*sigma_1(n) + (5*n/24)*sigma_3(n). - Ridouane Oudra, Sep 17 2020
Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 2160. - Vaclav Kotesovec, May 09 2022

More terms from Sean A. Irvine, Nov 14 2010

a(1)=0 prepended by Michel Marcus, Feb 02 2014

A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

Original entry on oeis.org

1, 8, 22, 57, 91, 185, 247, 402, 545, 775, 917, 1379, 1573, 1995, 2455, 3106, 3428, 4377, 4775, 5909, 6753, 7727, 8301, 10331, 11230, 12564, 13904, 15990, 16888, 19908, 20930, 23597, 25545, 27767, 29827, 34468, 35910, 38660, 41328, 46318, 48080, 53644, 55578

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A064602, A143128, A222548, A350107, A350124, A350125, A350128.

Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)

PARI
```
a(n) = sum(k=1, n, k^2*(n\k)^2);
```

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));

a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

from math import isqrt
def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (2*d - 1)/d^2 = Sum_{k=1..n} 2 * k * sigma(k) - sigma_2(k) = 2 * A143128(n) - A064602(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) ~ n^3 * (Pi^2/9 - zeta(3)/3). - Vaclav Kotesovec, Dec 16 2021

A256533 Product of n and the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 8, 24, 60, 105, 198, 287, 448, 621, 870, 1089, 1524, 1833, 2310, 2835, 3520, 4046, 4986, 5643, 6780, 7791, 8954, 9913, 11784, 13050, 14664, 16308, 18480, 20010, 22860, 24614, 27424, 29865, 32606, 35245, 39528, 42032, 45448, 48828, 53680, 56744, 62160, 65532, 70752, 75870, 80868, 84882, 92640, 97363, 104000

Offset: 1

Omar E. Pol, May 02 2015

nonn

a(n) is also sum of the volumes (or the total number of unit cubes) from two complementary polycubes: the irregular staircase after n-th stage described in A244580, and the irregular stepped pyramid after (n-1)st stage described in A245092. Note that in both structures the horizontal area in the n-th level is also the symmetric representation of sigma(n). This comment is represented by the third formula.

			For n = 3; a(3) = 3 * 8 = 19 + 5 = 24.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A024916, A064987, A143128, A175254, A196020, A236104, A237270, A237271, A237593, A244580, A245092, A245100, A256532, A332264.

a[n_]:=n*Apply[Plus,Flatten[Divisors[Range[n]]]]; Array[a,50] (* Ivan N. Ianakiev, May 03 2015 *)
nxt[{n_,sd_,a_}]:=Module[{k=(n+1)*(DivisorSigma[1,n+1]+sd)},{n+1,sd+DivisorSigma[ 1,n+1],k}]; NestList[ nxt,{1,1,1},50][[;;,3]] (* Harvey P. Dale, Jun 12 2023 *)

a(n) = n*sum(k=1, n, n\k*k); \\ Michel Marcus, Apr 29 2020

def A256533(n):
    s=0
    for k in range(1, n+1):
        s+=n%k
    return (n**3)-(s*n) # Indranil Ghosh, Feb 13 2017

from math import isqrt
def A256533(n): return n*(-(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 22 2023

a(n) = n*A024916(n).
a(n) = n^3 - A256532(n).
a(n) = A143128(n) + A175254(n-1), n > 1.
a(n) = A332264(n) + A175254(n). - Omar E. Pol, Apr 29 2020

A319086 a(n) = Sum_{k=1..n} k^2*sigma(k), where sigma is A000203.

Original entry on oeis.org

1, 13, 49, 161, 311, 743, 1135, 2095, 3148, 4948, 6400, 10432, 12798, 17502, 22902, 30838, 36040, 48676, 55896, 72696, 86808, 104232, 116928, 151488, 170863, 199255, 228415, 272319, 297549, 362349, 393101, 457613, 509885, 572309, 631109, 749045, 801067

Offset: 1

Vaclav Kotesovec, Sep 10 2018

nonn

In general, for m>=1, Sum_{k=1..n} k^m * sigma(k) = Sum_{k=1..n} k^(m+1) * (Bernoulli(m+1, floor(1 + n/k)) - Bernoulli(m+1, 0)) / (m+1), where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 08 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A024916, A143128.

Accumulate[Table[k^2*DivisorSigma[1, k], {k, 1, 50}]]

a(n) = sum(k=1, n, k^2*sigma(k)); \\ Michel Marcus, Sep 12 2018

def A319086(n): return sum((k*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A319086(n): return ((-((s:=isqrt(n))*(s+1))**3*(2*s+1)>>1) + sum(k**2*(q:=n//k)*(q+1)*(2*k*(2*q+1)+3*q*(q+1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) ~ Pi^2 * n^4/24.

a(n) = Sum_{k=1..n} ((k/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018

A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).

Original entry on oeis.org

1, 1, 4, 1, 5, 8, 1, 7, 11, 15, 1, 11, 19, 23, 21, 1, 19, 41, 47, 33, 33, 1, 35, 103, 125, 77, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99

Offset: 1

Seiichi Manyama, Jul 27 2022

			Square array begins:
   1,  1,   1,   1,    1,     1,     1, ...
   4,  5,   7,  11,   19,    35,    67, ...
   8, 11,  19,  41,  103,   281,   799, ...
  15, 23,  47, 125,  395,  1373,  5027, ...
  21, 33,  77, 255, 1025,  4503, 20657, ...
  33, 57, 149, 555, 2537, 12867, 68969, ...

Column k=0..4 give A024916, A143127, A143128, A356125, A356126.
T(n,n) gives A356129.
T(n,n+1) gives A356128.
Cf. A279394, A319649, A350106.

T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 28 2022 *)

T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));

T(n, k) = sum(j=1, n, j*sigma(j, k-1));

from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1
def A356124_gen(): # generator of terms
     return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n))
A356124_list = list(islice(A356124_gen(),30)) # Chai Wah Wu, Oct 24 2023

G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.

T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).

A256532 Product of n and the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

Original entry on oeis.org

0, 0, 3, 4, 20, 18, 56, 64, 108, 130, 242, 204, 364, 434, 540, 576, 867, 846, 1216, 1220, 1470, 1694, 2254, 2040, 2575, 2912, 3375, 3472, 4379, 4140, 5177, 5344, 6072, 6698, 7630, 7128, 8621, 9424, 10491, 10320, 12177, 11928, 13975, 14432, 15255, 16468, 18941, 17952, 20286, 21000, 22899, 23608, 26765, 26568, 29095

Offset: 1

Omar E. Pol, May 03 2015

nonn

a(n) is also the volume (or the total number of unit cubes) of a polycube which is a right prism whose base is the symmetric representation of A004125(n).

Note that the union of this right prism and the irregular staircase after n-th stage described in A244580 and the irregular stepped pyramid after (n-1)-th stage described in A245092, form a hexahedron (or cube) of side length n. This comment is represented by the third formula.

			a(5) = 20 because 5 * (0 + 1 + 2 + 1) = 5 * 4 = 20.
a(6) = 18 because 6 * (0 + 0 + 0 + 2 + 1) = 6 * 3 = 18.
a(7) = 56 because 7 * (0 + 1 + 1 + 3 + 2 + 1) = 7 * 8 = 56.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A004125, A024916, A143128, A175254, A236104, A236112, A237270, A237271, A237593, A244580, A245092, A256533.

Table[n*Sum[Mod[n,i],{i,2,n-1}],{n,55}] (* Ivan N. Ianakiev, May 04 2015 *)

vector(50, n, n*sum(k=1, n, n % k)) \\ Michel Marcus, May 05 2015

def A256532(n):
    s=0
    for k in range(1,n+1):
        s+=n%k
    return s*n # Indranil Ghosh, Feb 13 2017

from math import isqrt
def A256532(n): return n**3+n*((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 22 2023

a(n) = n * A004125(n).
a(n) = n^3 - A256533(n).
a(n) = n^3 - A143128(n) - A175254(n-1), n > 1.

A332264 Partial sums of A334136.

Original entry on oeis.org

0, 3, 11, 32, 56, 116, 164, 269, 373, 535, 655, 963, 1131, 1443, 1779, 2244, 2532, 3195, 3555, 4353, 4993, 5749, 6277, 7657, 8401, 9451, 10491, 12003, 12843, 14931, 15891, 17844, 19380, 21162, 22794, 25979, 27347, 29567, 31695, 35205, 36885, 40821, 42669, 46281, 49713, 52953, 55161, 60989, 63725, 68282

Offset: 1

Omar E. Pol, Apr 19 2020

nonn

a(n) is also the volume after n-th step of the symmetric staircase described in A244580 except the volume of the base level.

			For n = 4 the volume of the first four levels of the symmetric staircase described in A244580 is 47, since the structure contains 47 cubes. The volume of the base level is 15, since the base of the structure contains 15 cubes, so a(4) = 47 - 15 = 32.

Cf. A000203, A024916, A064987, A143128, A175254, A237593, A244580, A256533, A334136.

a(n) = sum(k=1, n, (k-1)*sigma(k)); \\ Michel Marcus, Apr 19 2020

from math import isqrt
def A332264(n): return (((s:=isqrt(n))**2*(s+1)*(6-(s+1)*((s<<1)+1))>>1)+sum((q:=n//k)*(k*(q+1)*(3*k+(q<<1)+1)-3*((k<<1)+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 25 2023

a(n) = A143128(n) - A024916(n).

a(n) = A256533(n) - A175254(n). - Omar E. Pol, Apr 29 2020

cp's OEIS Frontend

A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

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A244580 Square array read by antidiagonals related to the symmetric representation of sigma.

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A340793 Sequence whose partial sums give A000203.

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A000441 a(n) = Sum_{k=1..n-1} k*sigma(k)*sigma(n-k).

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A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

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A256533 Product of n and the sum of all divisors of all positive integers <= n.

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A319086 a(n) = Sum_{k=1..n} k^2*sigma(k), where sigma is A000203.

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A000441 a(n) = Sum_{k=1..n-1} ksigma(k)sigma(n-k).