A007429 -id:A007429 - cp's OEIS frontend

A069914 a(n) = Sum_{d|n} (d-1)*sigma(n/d).

0, 1, 2, 6, 4, 15, 6, 23, 16, 27, 10, 64, 12, 39, 42, 72, 16, 98, 18, 110, 60, 63, 22, 213, 48, 75, 84, 156, 28, 245, 30, 201, 96, 99, 102, 380, 36, 111, 114, 357, 40, 345, 42, 248, 248, 135, 46, 618, 96, 278, 150, 294, 52, 478, 162, 501, 168, 171, 58, 924, 60, 183

Offset: 1

Vladeta Jovovic, May 04 2002

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A006579, A007429, A060640.

Table[Plus @@ (DivisorSigma[1, ds = Divisors[n]]*(n/ds - 1)), {n, 62}] (* Ivan Neretin, May 17 2015 *)

a(n) = sumdiv(n, d, (d-1)*sigma(n/d)) \\ Michel Marcus, Jun 17 2013

a(n) = A060640(n) - A007429(n).

G.f.: Sum_{k>=1} sigma(k) * x^(2*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Aug 19 2021

A076663 a(n) = sum of sigma(e) where e ranges over all non-divisors of n that are between 1 and n.

Original entry on oeis.org

0, 0, 3, 4, 14, 13, 32, 30, 51, 59, 86, 72, 126, 129, 154, 163, 219, 205, 276, 262, 326, 355, 406, 361, 484, 504, 546, 561, 659, 622, 761, 737, 840, 883, 944, 900, 1097, 1112, 1177, 1160, 1341, 1300, 1479, 1465, 1560, 1658, 1757, 1645, 1921, 1928, 2057, 2085

Offset: 1

Joseph L. Pe, Oct 24 2002

nonn

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

Cf. A000203, A007429, A024916.

N:= 1000: # for a(1) .. a(N)
S:= [seq(numtheory:-sigma(n),n=1..N)]:
SS:= ListTools:-PartialSums(S):
seq(SS[n] - add(S[d], d = numtheory:-divisors(n)), n=1..N); # Robert Israel, Oct 27 2024

f[n_] := Module[{s, i}, s = 0; For[i = 1, i < n, i++, If[Mod[n, i] != 0, s = s + DivisorSigma[1, i]]]; s]; Table[f[i], {i, 1, 100}]

a(n) = A024916(n) - A007429(n). - Robert Israel, Oct 27 2024

A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

Original entry on oeis.org

1, 4, 1, 5, 1, 1, 11, 4, 1, 1, 7, 1, 1, 1, 1, 20, 8, 4, 1, 1, 19, 1, 1, 1, 1, 1, 1, 26, 11, 4, 4, 1, 1, 1, 1, 18, 5, 5, 1, 1, 1, 1, 1, 1, 28, 10, 4, 4, 4, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 55, 27, 15, 8, 4, 4, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 06 2008

Left border = A007429: (1, 4, 5, 11, 7, 20, 9,...).

Row sums = A060640: (1, 5, 7, 17, 11, 35,...).

			First few rows of the triangle =
1;
4, 1;
5, 1, 1;
11, 4, 1, 1;
7, 1, 1, 1, 1;
20, 8, 4, 1, 1, 1;
9, 1, 1, 1, 1, 1, 1;
...
Row 4 = (11, 4, 1, 1) since row 4 of A130540 = (7, 3, 0, 1).

Cf. A130540, A007429, A060640.

Triangle read by rows, A130540 * A000012, 1<=k<=n. Equals partial row sums of A130540 starting from the right.

A206029 a(n) = sum of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

0, 2, 5, 17, 14, 58, 27, 94, 73, 143, 65, 351, 90, 264, 265, 439, 152, 708, 189, 826, 483, 614, 275, 1700, 458, 843, 762, 1497, 434, 2488, 495, 1896, 1111, 1409, 1113, 3988, 702, 1746, 1521, 3913, 860, 4476, 945, 3427, 2955, 2528, 1127, 7465, 1587, 4219

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

In sequence A007429 are added all values of sigma(d) of all divisors d of numbers n, in sequence A206028 are added only distinct values of sigma(d) of all divisors d of numbers n and in sequence a(n) are added numbers k (1<=k<=sigma(n)) such that sigma(d) = k has no solution for neither divisor d of number n.

			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 = 2+5+6+7+8+9+10+11 = 58.

Cf. A000203, A206030, A007429, A206028.

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

a(n) = A184387(n) - A206028 = A000217(A000203(n)) - A206028.

A326826 a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.

Original entry on oeis.org

1, 5, 8, 19, 17, 43, 30, 69, 60, 95, 68, 176, 93, 171, 166, 255, 155, 342, 192, 403, 303, 395, 278, 681, 358, 543, 490, 738, 437, 961, 498, 969, 709, 911, 720, 1476, 705, 1131, 978, 1603, 863, 1773, 948, 1732, 1440, 1643, 1130, 2634, 1284, 2110, 1648, 2391, 1433, 2882, 1706

Offset: 1

Ilya Gutkovskiy, Oct 20 2019

nonn

Inverse Moebius transform applied twice to triangular numbers (A000217).

Cf. A000005, A000203, A000217, A001157, A007429, A007433, A007437, A092346, A326828.

[(1/2)*&+[DivisorSigma(1,d)+DivisorSigma(2,d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 20 2019

with(numtheory):
a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019

Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ Michel Marcus, Oct 20 2019

G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j) / (1 - x^(i*j))^3.
G.f.: (1/2) * Sum_{i>=1} Sum_{j>=1} j * (j + 1) * x^(i*j) / (1 - x^(i*j)).
G.f.: (1/2) * Sum_{k>=1} (sigma_1(k) + sigma_2(k)) * x^k / (1 - x^k).
Dirichlet g.f.: zeta(s)^2 * (zeta(s-1) + zeta(s-2)) / 2.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * tau(n/d), where tau = A000005.
a(n) = Sum_{d|n} A007437(d).
Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Dec 11 2021

A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.

Original entry on oeis.org

1, 0, 5, -5, 7, 0, 9, -18, 18, 0, 13, -25, 15, 0, 35, -47, 19, 0, 21, -35, 45, 0, 25, -90, 38, 0, 58, -45, 31, 0, 33, -108, 65, 0, 63, -90, 39, 0, 75, -126, 43, 0, 45, -65, 126, 0, 49, -235, 66, 0, 95, -75, 55, 0, 91, -162, 105, 0, 61, -175, 63, 0, 162, -233, 105

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

Inverse Moebius transform of A002129.

Dirichlet convolution of A000005 with A181983.

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

Cf. A000005, A002129, A007429, A008586 (positions of negative terms), A016825 (positions of 0's), A181983, A288417, A288571.

[&+[(-1)^(d+1)*d*#Divisors(n div d):d in Divisors(n)]:n in [1..65]]; // Marius A. Burtea, Sep 14 2019

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[(-1)^(d + 1) d DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := (p^(e + 2) - (e + 2)*p + e + 1)/(p-1)^2; f[2, e_] := 3*e + 5 - 2^(e+2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, May 25 2025 *)

a(n) = {sumdiv(n, d, (-1)^(d + 1) * d * numdiv(n/d))} \\ Andrew Howroyd, Sep 14 2019

a(n) = Sum_{d|n} A002129(d).
a(n) = Sum_{d|n} (-1)^(d + 1) * d * tau(n/d).
Multiplicative with a(2^e) = 3*e + 5 - 2^(e+2), and a(p^e) = (p^(e+2) - (e+2)*p +e + 1)/(p-1)^2 for an odd prime p. - Amiram Eldar, May 25 2025

A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 7, 31, 287, 3131, 47527, 823551, 16843583, 387440266, 10009772937, 285311670623, 8918294639219, 302875106592267, 11112685050294387, 437893920912795941, 18447025553014982271, 827240261886336764195, 39346558271492953948522, 1978419655660313589123999

Offset: 1

Wesley Ivan Hurt, May 28 2021

nonn

If p is prime, a(p) = p * Sum_{d|p} sigma_d(d) / d = p * (1 + (1^p + p^p)/p) = 1 + p + p^p.

			a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A245466, A321141, A334874, A343781, A344434, A344480.

Cf. A001001, A007429, A027847, A027848, A060640.

Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Dec 16 2022

G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k)^2. - Seiichi Manyama, Dec 16 2022

A066365 f-perfect numbers, where f(m) = sigma(m)-m.

Original entry on oeis.org

1, 1134, 1476, 1530, 16600, 282555

Offset: 1

Joseph L. Pe, Dec 21 2001

f-perfect numbers for an arithmetical function f is defined in A066218.

Also, numbers m such that 3*sigma(m)-2*m = A007429(m). - Max Alekseyev, Jul 30 2025

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A007429, A066218.

f[x_] := Abs[DivisorSigma[1, x] - x]; Select[ Range[2, 10^6], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

Edited and a(1)=1 inserted by Max Alekseyev, Jul 30 2025

A069913 a(n) = Sum_{d|n} (d-1)*tau(n/d).

Original entry on oeis.org

0, 1, 2, 5, 4, 11, 6, 16, 12, 19, 10, 37, 12, 27, 26, 42, 16, 54, 18, 59, 36, 43, 22, 100, 32, 51, 48, 81, 28, 113, 30, 99, 56, 67, 54, 162, 36, 75, 66, 152, 40, 153, 42, 125, 108, 91, 46, 240, 60, 134, 86, 147, 52, 202, 82, 204, 96, 115, 58, 331, 60, 123, 144, 219, 96

Offset: 1

Vladeta Jovovic, May 04 2002

nonn

Cf. A000005, A006579, A007425, A007429.

Table[DivisorSum[n,(#-1)DivisorSigma[0,n/#]&],{n,100}] (* Giorgos Kalogeropoulos, Aug 19 2021 *)

a(n) = sumdiv(n, d, (d-1)*numdiv(n/d)) \\ Michel Marcus, Jun 17 2013

a(n) = A007429(n) - A007425(n).

G.f.: Sum_{k>=1} tau(k) * x^(2*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Aug 19 2021

A127570 Triangle T(n,k) = sigma(k) if k|n, otherwise T(n,k)=0; 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jan 19 2007

			First few rows of the triangle are:
  1;
  1, 3;
  1, 0, 4;
  1, 3, 0, 7;
  1, 0, 0, 0, 6;
  1, 3, 4, 0, 0, 12;
  ...

Cf. A000203 (sigma, diagonal n=k), A007429 (row sums), A051731.

T(n,k) = Sum_{j=k..n} A051731(n,j)*A130208(j,k) = A051731(n,k)*A000203(k).

cp's OEIS Frontend

A069914 a(n) = Sum_{d|n} (d-1)*sigma(n/d).

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A076663 a(n) = sum of sigma(e) where e ranges over all non-divisors of n that are between 1 and n.

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A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

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A206029 a(n) = sum of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

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A326826 a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.

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A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.

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Magma

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A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

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A066365 f-perfect numbers, where f(m) = sigma(m)-m.

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