A007429 -id:A007429 - cp's OEIS frontend

A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  3,   4,   6,   10,    18,    34,  ...
  3,   5,  11,   29,    83,   245,  ...
  6,  11,  27,   83,   291,  1091,  ...
  3,   7,  27,  127,   627,  3127,  ...
  9,  20,  66,  290,  1494,  8330,  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..3 give A007425, A007429, A007433, A321140.

Cf. A109974, A321141 (diagonal), A356045.

Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, d^k*numdiv(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

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

A(n,k) = Sum_{d|n} d^k*tau(n/d).

A322655 Numerator of (Sum_{d|n} sigma(d)) / sigma(n).

Original entry on oeis.org

1, 4, 5, 11, 7, 5, 9, 26, 18, 14, 13, 55, 15, 3, 35, 57, 19, 24, 21, 11, 45, 13, 25, 13, 38, 10, 29, 99, 31, 35, 33, 40, 65, 38, 21, 198, 39, 7, 75, 91, 43, 15, 45, 143, 21, 25, 49, 285, 22, 152, 95, 165, 55, 29, 91, 39, 21, 62, 61, 55, 63, 11, 81, 247, 5, 65

Offset: 1

Jaroslav Krizek, Dec 22 2018

Numerator of A007429(n) / A000203(n).

Also numerator of Sum_{d|n} (sigma(d) / sigma(n)).

			For n = 4; a(4) = numerator((Sum_{d|4} sigma(d)) / sigma(4)) = numerator((1 + 3 + 7) / (1 + 2 + 4)) = numerator(11/7) = 11.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A007429, A319296, A221219, A322656 (denominator).

[Numerator(&+[SumOfDivisors(d): d in Divisors(n)] /  SumOfDivisors(n)): n in [1..1000]];

Table[Numerator[Sum[DivisorSigma[1, d], {d, Divisors[n]}] / DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 22 2018 *)

a(n) = numerator(sumdiv(n, d, sigma(d))/sigma(n)); \\ Michel Marcus, Dec 22 2018

a(n) = A007429(n) / gcd(A000203(n), A007429(n)). - Antti Karttunen, Nov 15 2021

A322656 Denominator of (Sum_{d|n} sigma(d)) / sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 15, 13, 9, 12, 28, 14, 2, 24, 31, 18, 13, 20, 6, 32, 9, 24, 6, 31, 7, 20, 56, 30, 18, 32, 21, 48, 27, 16, 91, 38, 5, 56, 45, 42, 8, 44, 84, 13, 18, 48, 124, 19, 93, 72, 98, 54, 15, 72, 20, 16, 45, 60, 24, 62, 8, 52, 127, 4, 36, 68, 126, 96

Offset: 1

Jaroslav Krizek, Dec 22 2018

Denominator of A007429(n) / A000203(n).

Also denominator of Sum_{d|n} (sigma(d) / sigma(n)).

			For n = 4; a(4) = denominator((Sum_{d|4} sigma(d)) / sigma(4)) = denominator((1 + 3 + 7) / (1 + 2 + 4)) = denominator(11/7) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A007429, A319296, A221219, A322655 (numerator).

[Denominator(&+[SumOfDivisors(d): d in Divisors(n)] /  SumOfDivisors(n)): n in [1..1000]];

Table[Denominator[Sum[DivisorSigma[1, d], {d, Divisors[n]}] / DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 22 2018 *)

a(n) = denominator(sumdiv(n, d, sigma(d))/sigma(n)); \\ Michel Marcus, Dec 22 2018

a(n) = 1 for numbers in A221219.

a(n) = A000203(n) / gcd(A000203(n), A007429(n)). - Antti Karttunen, Nov 15 2021

A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

Original entry on oeis.org

1, 4, 5, 9, 7, 20, 9, 18, 15, 28, 13, 45, 15, 36, 35, 35, 19, 60, 21, 63, 45, 52, 25, 90, 33, 60, 43, 81, 31, 140, 33, 68, 65, 76, 63, 135, 39, 84, 75, 126, 43, 180, 45, 117, 105, 100, 49, 175, 59, 132, 95, 135, 55, 172, 91, 162, 105, 124, 61, 315, 63, 132, 135, 133, 105

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Inverse Moebius transform of A034448.

Dirichlet convolution of A055615 with A064840.

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

Cf. A000005, A000203, A007429, A008683, A034448, A048691, A055615, A064840, A319131, A319132.

with(numtheory):
a:= n-> add(mobius(d)*tau(n/d)*sigma(n/d)*d, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 16 2019

Table[n DivisorSum[n, MoebiusMu[n/#] DivisorSigma[0, #] DivisorSigma[1, #]/# &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSum[k, # &, CoprimeQ[#, k/#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^(e + 1) - p)/(p - 1) + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2023 *)

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, (p[i]^(e[i] + 1) - p[i])/(p[i] - 1) + e[i] + 1);} \\ Amiram Eldar, Feb 10 2023

G.f.: Sum_{k>=1} usigma(k) * x^k / (1 - x^k), where usigma = A034448.
a(n) = Sum_{d|n} usigma(d).
a(n) = n * Sum_{d|n} mu(n/d) * tau(d) * sigma(d) / d, where mu = A008683, tau = A000005 and sigma = A000203.
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (72 * zeta(3)). - Vaclav Kotesovec, Oct 17 2019
From Amiram Eldar, Feb 10 2023: (Start)
a(n) = Sum_{d|n} Sum_{d'|n, gcd(d, d')=1} d'.
Multiplicative with a(p^e) = (p^(e+1)-p)/(p-1) + e + 1. (End)

A344060 a(n) = Sum_{d|n} sigma(d)^n.

Original entry on oeis.org

1, 10, 65, 2483, 7777, 2990810, 2097153, 2568661988, 10604761518, 3570527751850, 743008370689, 232227195048256531, 793714773254145, 21035724521219881850, 504857283427304833025, 727429690188773950335429, 2185911559738696531969, 43567528891100073055151954340, 5242880000000000000000001

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A023887, A065018, A342471, A344044, A344047, A344061.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^n &]; Array[a, 19] (* Amiram Eldar, May 08 2021 *)

PARI
```
a(n) = sumdiv(n, d, sigma(d)^n);
```

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (sigma(k)*x)^k/(1-(sigma(k)*x)^k)))

G.f.: Sum_{k >= 1} (sigma(k) * x)^k/(1 - (sigma(k) * x)^k).

If p is prime, a(p) = 1 + (p+1)^p.

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 4, 0, 1, 7, 3, 0, 1, 6, 0, 0, 0, 1, 12, 4, 3, 0, 0, 1, 8, 0, 0, 0, 0, 0, 1, 15, 7, 0, 3, 0, 0, 0, 1, 13, 0, 4, 0, 0, 0, 0, 0, 1, 18, 6, 0, 0, 3, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 03 2007

The original definition was: A127093 * A125093^(-1).
Left border = A000203, sigma(n): (1, 3, 4, 7, 6, ...). Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, ...); = inverse Moebius transform applied to sigma(n); (i.e., inverse Moebius transform applied twice to natural numbers).
T(n,k) is the total number of parts congruent to 0 mod k in the partitions of n into equal parts. - Omar E. Pol, Nov 19 2019
From Omar E. Pol, Jan 01 2020: (Start)
Conjecture 1: the sum of odd-indexed terms in row n equals A327096(n).
Conjecture 2: the sum of even-indexed terms in row n equals the n-th term of the sequence formed by A000004 and A007429 interleaved.
Conjecture 3: alternating row sums give A288417. (End)

			First few rows of the triangle are:
   1;
   3,  1;
   4,  0, 1;
   7,  3, 0, 1;
   6,  0, 0, 0, 1;
  12,  4, 3, 0, 0, 1;
   8,  0, 0, 0, 0, 0, 1;
  15,  7, 0, 3, 0, 0, 0, 1;
  13,  0, 4, 0, 0, 0, 0, 0, 1;
  18,  6, 0, 0, 3, 0, 0, 0, 0, 1;
  12,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1;
  14,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  24,  8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1;
  24,  0, 6, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  31, 15, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1;
...
Extended by _Omar E. Pol_, Nov 19 2019

Cf. A000203, A007429, A127093, A125093, A244051, A288417, A327096.

A127093 * A125093^(-1), as infinite lower triangular matrices.

New name and more terms from Omar E. Pol, Nov 19 2019

A134699 Triangle read by rows: A051731^2 * A000012.

Original entry on oeis.org

1, 3, 1, 3, 1, 1, 6, 3, 1, 1, 3, 1, 1, 1, 1, 9, 5, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 6, 3, 3, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 1, 1, 9, 5, 3, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 12, 8, 5, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 06 2007

Left column = A007425.
Row sums = A007429: (1, 4, 5, 11, 7, 20, ...).
A134700 = A000012 * A127170.

			First few rows of the triangle:
   1;
   3, 1;
   3, 1, 1;
   6, 3, 1, 1;
   3, 1, 1, 1, 1;
   9, 5, 3, 1, 1, 1;
   3, 1, 1, 1, 1, 1, 1;
  10, 6, 3, 3, 1, 1, 1, 1;
  ...

Cf. A007425, A007429, A051731, A127170, A134700.

A051731^2 * A000012 = A127170 * A000012, as infinite lower triangular matrices.

More terms from Jinyuan Wang, Apr 29 2025

A145378 a(n) = Sum_{d|n} sigma(d) - 2Sum_{2c|n} sigma(c) + 4Sum_{4b|n} sigma(b).

Original entry on oeis.org

1, 2, 5, 7, 7, 10, 9, 20, 18, 14, 13, 35, 15, 18, 35, 49, 19, 36, 21, 49, 45, 26, 25, 100, 38, 30, 58, 63, 31, 70, 33, 110, 65, 38, 63, 126, 39, 42, 75, 140, 43, 90, 45, 91, 126, 50, 49, 245, 66, 76, 95, 105, 55, 116, 91, 180, 105, 62, 61, 245, 63, 66, 162, 235, 105, 130, 69

Offset: 1

N. J. A. Sloane, Mar 12 2009

Dirichlet convolution of [1,-2,0,4,0,0,0,...] with A007429.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. See g(n).

Cf. A007429, A300707.

with(numtheory); g:=proc(n) local d,c,b,t0,t1,t2,t3;
t1:=divisors(n);
t0:=add( sigma(d), d in t1);
t2:=0; for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
t3:=0; for d in t1 do if d mod 4 = 0 then t3:=t3+sigma(d/4); fi; od:
t0-2*t2+4*t3; end;
[seq(g(n),n=1..100)];
# alternative
read("transforms") : nmax := 100 :
L27 := [seq(i,i=1..nmax) ];
L := [1,-2,0,4,seq(0,i=1..nmax)] ;
DIRICHLET(L27,L) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

a[n_] := Sum[DivisorSigma[1, d] - 2 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2] + 4 Boole[Mod[d, 4] == 0] DivisorSigma[1, d/4], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Apr 04 2020 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 2^(e + 2) - 3*(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2^(f[i,2]+2) - 3*(f[i,2]+1),  (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022

Dirichlet g.f.: (1-2/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
From Amiram Eldar, Oct 25 2022: (Start)
Multiplicative with a(2^e) = 2^(e+2) - 3*(e+1) and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). (End)

A145396 a(n) = Sum_{d|n} sigma(d) + 3*Sum_{2c|n} sigma(c).

Original entry on oeis.org

1, 7, 5, 23, 7, 35, 9, 59, 18, 49, 13, 115, 15, 63, 35, 135, 19, 126, 21, 161, 45, 91, 25, 295, 38, 105, 58, 207, 31, 245, 33, 291, 65, 133, 63, 414, 39, 147, 75, 413, 43, 315, 45, 299, 126, 175, 49, 675, 66, 266, 95, 345, 55, 406, 91, 531, 105, 217, 61, 805, 63, 231, 162, 607

Offset: 1

N. J. A. Sloane, Mar 13 2009

Dirichlet convolution of [1,3,0,0,0,0,0,...] and A007429.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. See Table 1, symmetry P2/m.

Cf. A007429.

with(numtheory);
g:=proc(n)
local d,c,b,t0,t1,t2,t3;
t1:=divisors(n);
t0:=add( sigma(d), d in t1);
t2:=0;
for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
t0+3*t2;
end;
[seq(g(n),n=1..100)];
# alternative
nmax := 100 :
L27 := [seq(i,i=1..nmax) ];
L := [1,3,seq(0,i=1..nmax)] ;
MOBIUSi(%) ;
MOBIUSi(%) ;
DIRICHLET(%,L27) ; # R. J. Mathar, Sep 25 2017

a[n_] := Sum[DivisorSigma[1, d] + 3 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Apr 04 2020 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 5*2^(e + 1) - 4*e - 9; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 5*2^(f[i,2]+1) - 4*f[i,2] - 9,  (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022

Dirichlet g.f.: (1+3/2^s)*zeta(s-1)*(zeta(s))^2.
From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2^e) = 5*2^(e+1)-4*e-9, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*Pi^4/288 = 2.367582... . (End)

A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

Original entry on oeis.org

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 248

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A007429: a(66) = 248, A007429(66) = 260. Number 66 is the smallest number with at least two divisors d with the same sigma(d); see A206030.
In A007429 all values of sigma(d) of the divisors d of n are included in the sum with repetitions allowed. In this sequence only the distinct values of sigma(d) of the divisors d of n are included in the sum.
If a term is a prime p when n = 2^j then p = 2^(j+2)-(j+3) is also a term of A099440 (primes of the form 2^n-n-1). Greater of twin primes are terms. - Metin Sariyar, Apr 03 2020

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Sum of k = 1+3+4+12 = 20.
For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Sum of k = 1+3+4+12+36+48+144 = 248 (note that only one twelve is added.).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000203, A000217, A007429, A184387, A206029, A206030.

Table[Total[Union[DivisorSigma[1, Divisors[n]]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)={vecsum(Set(apply(sigma, divisors(n))))} \\ Andrew Howroyd, Aug 01 2018

a(p) = p+2, a(pq) = (p+2)*(q+2) for p, q = distinct primes.
a(n) = A184387(n) - A206029(n) = A000217(A000203(n)) - A206029(n).
a(2^n) = 2^(n+2) - (n+3). - Metin Sariyar, Apr 09 2020

Name clarified by David A. Corneth, Aug 01 2018

a(62)-a(66) from Andrew Howroyd, Aug 01 2018

cp's OEIS Frontend

A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

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A322655 Numerator of (Sum_{d|n} sigma(d)) / sigma(n).

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Magma

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A322656 Denominator of (Sum_{d|n} sigma(d)) / sigma(n).

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Magma

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A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

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Maple

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A344060 a(n) = Sum_{d|n} sigma(d)^n.

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A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

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A134699 Triangle read by rows: A051731^2 * A000012.

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A145378 a(n) = Sum_{d|n} sigma(d) - 2*Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

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A145378 a(n) = Sum_{d|n} sigma(d) - 2Sum_{2c|n} sigma(c) + 4Sum_{4b|n} sigma(b).