A023887 -id:A023887 - cp's OEIS frontend

A067313 Numbers k such that sigma_k(k)/k is an integer, where sigma_k(k) is the sum of the k-th powers of the divisors of k (A023887).

1, 10, 130, 135, 147, 150, 228, 250, 350, 364, 410, 444, 492, 876, 891, 945, 1014, 1308, 1372, 1550, 1690, 1950, 2050, 2210, 2373, 2565, 2850, 3045, 3050, 3250, 3375, 3876, 4108, 4185, 4905, 4995, 5050, 5070, 5145, 5330, 5439, 5481, 6150, 6250, 6321, 6615, 6890, 7514

Offset: 1

Labos Elemer, Jan 14 2002

nonn

			If k = 10, then sigma(10,10) = 1 + 1024 + 9765625 + 10000000000 = 1009766650 is divisible by k = 10.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Amiram Eldar)

Cf. A023887.

Do[s=DivisorSigma[n, n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10000}]
Select[Range[6500],Divisible[DivisorSigma[#,#],#]&] (* Harvey P. Dale, Feb 06 2019 *)

isok(k) = !(sigma(k, k) % k); \\ Michel Marcus, Aug 10 2020

is(n) = {my(d = divisors(n), v = vecsum(vector(#d - 1, i, Mod(d[i], n)^n))); lift(v)==0} \\ David A. Corneth, Aug 10 2020

A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 2, 7, 10, 9, 1, 4, 6, 21, 28, 17, 1, 2, 12, 26, 73, 82, 33, 1, 4, 8, 50, 126, 273, 244, 65, 1, 3, 15, 50, 252, 626, 1057, 730, 129, 1, 4, 13, 85, 344, 1394, 3126, 4161, 2188, 257, 1, 2, 18, 91, 585, 2402, 8052, 15626, 16513, 6562, 513, 1

Offset: 0

Paul Barry, Jul 06 2005

Rows sums are A108639. Antidiagonal sums are A109976. Matrix inverse is A109977.
From Wolfdieter Lang, Jan 29 2016: (Start)
The sum of the (k-1)th power of the divisors of n, sigma_(k-1)(n), appears also as eigenvalue lambda(k, n) of the Hecke operators T_n, n a positive integer, acting on the normalized Eisenstein series E_k(q) = ((2*Pi*i)^k/((k-1)!*Zeta(k))*G_k(q) with even k >= 4 and q = 2*Pi*i*z, where z is from the upper half of the complex plane: T_n E_k = sigma_(k-1)(n)*E_k. These Eisenstein series are entire modular forms of weight k, and each E_k(q) is a simultaneous eigenform of the Hecke operators T_n, for every n >= 1.
This results from the Fourier coefficients of E_k(q) = Sum_{m>=0} E(k, m)*q^m, with E(k, 0) =1 and E(k, m) = ((2*Pi*i)^k / ((k-1)!*Zeta(k))* sigma_(k-1)(m) for m >= 1, together with the Fourier coefficients of T_n E_k. The eigenvalues lambda(n, k) = (Sum_{d | gcd(n,m)} d^{k-1}*E(k, m*n/d^2)) / E(k, m) for each m >= 0. For m=0 this becomes lambda(n, k) = sigma_(k-1)(n).
For Hecke operators, Fourier coefficients and simultaneous eigenforms see, e.g., the Koecher - Krieg reference, p. 207, eqs. (5) and (6) and p. 211, section 4, or the Apostol reference, p. 120, eq. (13), pp. 129 - 134. (End)

			Start of array:
  1,  2,  2,   3,   2,    4, ...
  1,  3,  4,   7,   6,   12, ...
  1,  5, 10,  21,  26,   50, ...
  1,  9, 28,  73, 126,  252, ...
  1, 17, 82, 273, 626, 1394, ...
  ...
The triangle T(m, k) with row offset 1 starts:
  m\k 0  1  2   3    4    5    6    7   8  9 ...
  1:  1
  2:  2  1
  3:  2  3  1
  4:  3  4  5   1
  5:  2  7 10   9    1
  6:  4  6 21  28   17    1
  7:  2 12 26  73   82   33    1
  8:  4  8 50 126  273  244   65    1
  9:  3 15 50 252  626 1057  730  129   1
  10: 4 13 85 344 1394 3126 4161 2188 257  1
  ... - _Wolfdieter Lang_, Jan 14 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 120, 129 - 134.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.

Alois P. Heinz, Antidiagonals k = 0..140, flattened

Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954 - A013972.
Columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660.
Row sums A108639.
Diagonals A082245, A023887.
Related sequences: A082771, A109976, A109977, A109978.

A109974:= func< n,k | DivisorSigma(k-1, n-k+1) >;
[A109974(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 18 2023

with(numtheory):
seq(seq(sigma[k](1+d-k), k=0..d), d=0..12);  # Alois P. Heinz, Feb 06 2013

rows=12; Flatten[Table[DivisorSigma[k-n, n], {k,1,rows}, {n,k,1,-1}]] (* Jean-François Alcover, Nov 15 2011 *)

def A109974(n,k): return sigma(n-k+1, k-1)
flatten([[A109974(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 18 2023

Regarded as a triangle, T(n, k) = if(k<=n, sigma(k-1, n-k+1), 0). - Franklin T. Adams-Watters, Jul 17 2006
If the row index (the index of the antidiagonal of the array) is taken as m with offset 1 the triangle is T(m, k) = sigma_k(m-k), 1 <= k+1 <= m, otherwise 0. - Wolfdieter Lang, Jan 14 2016
G.f. for the triangle with offset 1: G(x,y) = Sum_{j>=1} x^j/((1-x^j)*(1-j*x*y)). - Robert Israel, Jan 14 2016

A023881 Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

Original entry on oeis.org

1, 1, 3, 12, 82, 725, 8811, 128340, 2257687, 45658174, 1052672116, 27108596725, 772945749970, 24137251258926, 819742344728692, 30069017799172228, 1184889562926838573, 49914141857616862435

Offset: 0

Olivier Gérard

nonn

			G.f. = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 725*x^5 + 8811*x^6 + 128340*x^7 + 2257687*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Cf. A023882, A023887, A158952.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k)^(1/k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

seq(coeff(series(mul((1-k^k*x^k)^(-1/k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

nmax=30; CoefficientList[Series[Product[1/(1-k^k*x^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 31 2018 *)

{a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, sigma(k, k) * x^k / k, x * O(x^n))), n))} /* Michael Somos, Feb 15 2006 */

{a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1-k^k*x^k+x*O(x^n))^(-1/k)),n))} /* Paul D. Hanna */

G.f.: exp( Sum_{k>0} sigma_k(k) * x^k / k). - Michael Somos, Feb 15 2006
G.f.: Product_{n>=1} (1 - n^n*x^n)^(-1/n). - Paul D. Hanna, Mar 08 2011
a(n) ~ n^(n-1). - Vaclav Kotesovec, Oct 08 2016

A342628 a(n) = Sum_{d|n} d^(n-d).

Original entry on oeis.org

1, 2, 2, 6, 2, 45, 2, 322, 731, 3383, 2, 132901, 2, 827641, 10297068, 33570818, 2, 2578617270, 2, 44812807567, 678610493340, 285312719189, 2, 393061010002613, 95367431640627, 302875123369471, 150094917726535604, 569939345952661545, 2, 105474306078445349841, 2

Offset: 1

Seiichi Manyama, Mar 16 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..599

Cf. A023887, A062796, A308594, A342629.

a[n_] := DivisorSum[n, #^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)))

from sympy import divisors
def A342628(n): return sum(d**(n-d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

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

If p is prime, a(p) = 2.

A217872 a(n) = sigma(n)^n.

Original entry on oeis.org

1, 9, 64, 2401, 7776, 2985984, 2097152, 2562890625, 10604499373, 3570467226624, 743008370688, 232218265089212416, 793714773254144, 21035720123168587776, 504857282956046106624, 727423121747185263828481, 2185911559738696531968, 43567528752021332753202420081

Offset: 1

Paul D. Hanna, Nov 01 2012

nonn

Here sigma(n) = A000203(n) is the sum of the divisors of n.

Compare to A023887(n) = sigma(n,n).

			L.g.f.: L(x) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 +...
where exponentiation yields the g.f. of A156217:
exp(L(x)) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A000203, A023887, A156217, A215140, A215141.

Table[DivisorSigma[1, n]^n, {n, 1, 20}] (* Amiram Eldar, Nov 16 2020 *)

{a(n)=sigma(n)^n}
for(n=1,20,print1(a(n),", "))

Logarithmic derivative of A156217.
From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=1} 1/a(n) = A215140.
Sum_{n>=1} (-1)^(n+1)/a(n) = A215141. (End)

A294645 a(n) = Sum_{d|n} d^(n+1).

Original entry on oeis.org

1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973

Offset: 1

Seiichi Manyama, Nov 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Divisor Function

Column k=1 of A308504.

Cf. A023882, A023887, A082245, A158095, A292312.

Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

PARI
```
{a(n) = sigma(n, n+1)}
```

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019
a(n) ~ n^(n+1). - Vaclav Kotesovec, Oct 07 2020

A321141 a(n) = Sum_{d|n} sigma_n(d).

Original entry on oeis.org

1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981

Offset: 1

Ilya Gutkovskiy, Oct 28 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386
N. J. A. Sloane, Transforms

Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

[&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020

with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018

Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]

a(n) = sumdiv(n, d, sigma(d, n)); \\ Michel Marcus, Oct 28 2018

from sympy import divisor_sigma, divisors
def A321141(n):
    return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^n*tau(n/d).
a(n) ~ n^n. - Vaclav Kotesovec, Feb 16 2020

A082245 Sum of (n-1)-th powers of divisors of n.

Original entry on oeis.org

1, 3, 10, 73, 626, 8052, 117650, 2113665, 43053283, 1001953638, 25937424602, 743375541244, 23298085122482, 793811662272744, 29192932133689220, 1152956690052710401, 48661191875666868482, 2185928253847184914509

Offset: 1

Reinhard Zumkeller, May 22 2003

nonn

a(n) = t(n,n-1), t as defined in A082771;

a(1)=A000005(1), a(2)=A000203(2), a(3)=A001157(3), a(4)=A001158(4), a(5)=A001159(5), a(6)=A001160(6), a(7)=A013954(7), a(8)=A013955(8).

			a(6) = 1^5 + 2^5 + 3^5 + 6^5 = 1 + 32 + 243 + 7776 = 8052.

Seiichi Manyama, Table of n, a(n) for n = 1..387
M. Sugunamma, Certain results concerning sigma_k(n) and phi_k(n), Annales Polonici Mathematici, Vol. 8, No. 2 (1960), pp. 173-176.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A023887, A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958
Cf. A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970
Cf. A001113, A013971, A013972.

[DivisorSigma(n-1, n): n in [1..20]]; // G. C. Greubel, Nov 02 2018

Table[Total[Divisors[n]^(n-1)], {n,18}] (* T. D. Noe, Oct 25 2006 *)
Table[DivisorSigma[n-1,n], {n,1,20}] (* G. C. Greubel, Nov 02 2018 *)

a(n) = sigma(n, n-1); \\ Michel Marcus, Nov 07 2017

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(1/k^2))))) \\ Seiichi Manyama, Jun 23 2019

[sigma(n,(n-1))for n in range(1,19)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 02 2018
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 23 2019
Limit_{n->oo} a(n)/A023887(n-1) = e (A001113) (Sugunamma, 1960). - Amiram Eldar, Apr 15 2021

Corrected by T. D. Noe, Oct 25 2006

A338661 a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 289, 3126, 49036, 823544, 17040385, 387538588, 10048833246, 285311670612, 8929334253419, 302875106592254, 11116754387182648, 437894348359764856, 18448995959423107073, 827240261886336764178, 39347761059781438793815, 1978419655660313589123980

Offset: 1

Seiichi Manyama, Apr 22 2021

nonn

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

Cf. A023887, A157019, A157020, A324158, A324159, A339481, A339482, A339712, A343573.

a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 2, #-1] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)

a(n) = sumdiv(n, d, d^n*binomial(d+n/d-2, d-1));

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

G.f.: Sum_{k >= 1} (k * x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p^p.
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

A343573 a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205

Offset: 1

Seiichi Manyama, Apr 20 2021

nonn

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

Cf. A023887, A157019, A157020, A324158, A324159, A338661, A339481, A339482, A339712, A343567, A343574.

a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)

a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));

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

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

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

cp's OEIS Frontend

A067313 Numbers k such that sigma_k(k)/k is an integer, where sigma_k(k) is the sum of the k-th powers of the divisors of k (A023887).

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A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

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A023881 Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

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A342628 a(n) = Sum_{d|n} d^(n-d).

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A217872 a(n) = sigma(n)^n.

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A294645 a(n) = Sum_{d|n} d^(n+1).

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A321141 a(n) = Sum_{d|n} sigma_n(d).

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