A013954 -id:A013954 - cp's OEIS frontend

A013955 a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n.

1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121, 410338674, 617285253, 893871740, 1290094638, 1801914272, 2513845188, 3404825448, 4624699020, 6103593751, 8094558822, 10465138360

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 51.
Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 3 p. 308.
Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_8(z).
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013666, A013954-A013972, A017665-A017712, A087115.

[DivisorSigma(7,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

lst={};Do[AppendTo[lst,DivisorSigma[7,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[7,Range[30]] (* Harvey P. Dale, Dec 10 2016 *)

PARI
```
a(n)=if(n<1,0,sigma(n,7))
```

[sigma(n, 7) for n in range(1, 23)]  # Zerinvary Lajos, Jun 04 2009

Let sigma(p,n) be the sum of the p-th powers of the divisors of n. Then sigma(7,n) = sigma(3,n) + 120 sum(sigma(3,k) sigma(3,n-k),k=1..n-1) (Cf. A087115). - Eugene Salamin, Apr 29 2006 [Hurwitz Identity, Math. Werke I, 1-66, p. 50, last line. See, e.g., the Koecher-Krieg reference, p. 51, rewritten. - Wolfdieter Lang, Jan 20 2016]
G.f.: Sum_{k>=1} k^7*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^6)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s)*zeta(s-7).
Sum_{k=1..n} a(k) = zeta(8) * n^8 / 8 + O(n^9). (End)

A050999 Sum of squares of odd divisors of n.

Original entry on oeis.org

1, 1, 10, 1, 26, 10, 50, 1, 91, 26, 122, 10, 170, 50, 260, 1, 290, 91, 362, 26, 500, 122, 530, 10, 651, 170, 820, 50, 842, 260, 962, 1, 1220, 290, 1300, 91, 1370, 362, 1700, 26, 1682, 500, 1850, 122, 2366, 530, 2210, 10, 2451, 651, 2900, 170, 2810, 820, 3172, 50, 3620, 842, 3482

Offset: 1

Eric W. Weisstein

Denoted by Delta_2(n) in Glaisher 1907. - Michael Somos, May 17 2013

The sum of squares of even divisors of 2*k = 4*A001157(k), and the sum of squares of even divisors of 2*k-1 vanishes, for k >= 1. - Wolfdieter Lang, Jan 07 2017

			x + x^2 + 10*x^3 + x^4 + 26*x^5 + 10*x^6 + 50*x^7 + x^8 + 91*x^9 + 26*x^10 + ...

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011, eq. (3.74).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher

Cf. A051000 - A051002, A000593, A001227, A000203, A001157-A001160, A013954-A013972.

Glaisher's Delta_i (i=0..12): A001227, A000593, A050999, A051000, A051001, A051002, A321810, A321811, A321812, A321813, A321814, A321815, A321816

a050999 = sum . map (^ 2) . a182469_row
-- Reinhard Zumkeller, May 01 2012

a[n_] := 1/2*Sum[(1 - (-1)^d)*d^2, {d, Divisors[n]}]; Table[a[n], {n, 1, 59}] (* Jean-François Alcover, Oct 23 2012, from 2nd formula *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d^2, {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)
f[p_, e_] := If[p == 2, 1, (p^(2*e + 2) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 22 2020 *)
Table[Total[Select[Divisors[n],OddQ]^2],{n,80}] (* Harvey P. Dale, Jul 19 2024 *)

a(n)=sumdiv(n,d, if(d%2==1, d^2, 0 ) );  /* Joerg Arndt, Oct 07 2012 */

from sympy import divisor_sigma
def A050999(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),2)) # Chai Wah Wu, Jul 16 2022

From Vladeta Jovovic, Sep 10 2001: (Start)
Multiplicative with a(p^e) = 1 if p = 2, (p^(2e+2)-1)/(p^2-1) if p > 2.
a(n) = (1/2)*Sum_{d|n} (1-(-1)^d)*d^2.
a(2n) = sigma_2(2n) - 4*sigma_2(n), a(2n+1) = sigma_2(2n+1), where sigma_2(n) is sum of squares of divisors of n (A001157).
More generally, if b(n, k) is the sum of k-th powers of odd divisors of n then b(2n, k) = sigma_k(2n)-2^k*sigma_k(n), b(2n+1, k) = sigma_k(2n+1). b(n, k) is multiplicative with a(p^e) = 1 if p = 2, (p^(k*e+k)-1)/(p^k-1) if p > 2. (End)
G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1-(2^k-1)*x^m)/(1-x^(2*m)). - Vladeta Jovovic, Oct 19 2002
Dirichlet g.f. (1-2^(2-s))*zeta(s)*zeta(s-2). - R. J. Mathar, Apr 06 2011
Dirichlet convolution of A001157 with [1,-4,0,0,0,0...]. Dirichlet convolution of [1,-3,1,-3,1,-3,..] with A000290. Dirichlet convolution of [1,0,9,0,25,0,49,0,81,...] with A000012 (or A057427). - R. J. Mathar, Jun 28 2011
a(n) = sum(A182469(n,k)^2: k=1..A001227(n)). [Reinhard Zumkeller, May 01 2012]
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / 6. - Vaclav Kotesovec, Nov 09 2018
G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(2*n) + x^(4*n))/(1 - x^(2*n))^3. - Peter Bala, Dec 19 2021
Sum_{k=1..n} (-1)^(k+1) * a(k) ~ zeta(3) * n^3 / 8. - Vaclav Kotesovec, Aug 07 2022

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

A013959 a(n) = sigma_11(n), the sum of the 11th powers of the divisors of n.

Original entry on oeis.org

1, 2049, 177148, 4196353, 48828126, 362976252, 1977326744, 8594130945, 31381236757, 100048830174, 285311670612, 743375541244, 1792160394038, 4051542498456, 8649804864648, 17600780175361, 34271896307634, 64300154115093, 116490258898220, 204900053024478

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Related to congruence properties of the Ramanujan tau function since A000594(n) == a(n) (mod 691) = A046694(n). - Benoit Cloitre, Aug 28 2002

T. D. Noe, Table of n, a(n) for n=1..1000
Index entries for sequences related to sigma(n).

Cf. A000594, A027860, A046694.

Cf. A000203, A001157-A001160, A013670, A013954-A013972, A017665-A017712.

[DivisorSigma(11, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

Table[DivisorSigma[11, n], {n, 30}] (* Vincenzo Librandi, Sep 10 2016 *)

a(n)=sigma(n,11) \\ Charles R Greathouse IV, Apr 28 2011

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^11*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

from sympy import divisor_sigma
def A013959(n): return divisor_sigma(n,11) # Chai Wah Wu, Nov 17 2022

[sigma(n,11)for n in range(1,18)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^11*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-11)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(11*e+11)-1)/(p^11-1).
Sum_{k=1..n} a(k) = zeta(12) * n^12 / 12 + O(n^13). (End)

A013957 a(n) = sigma_9(n), the sum of the 9th powers of the divisors of n.

Original entry on oeis.org

1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, 118587876498, 198756808749, 322687697780, 513002215782, 794320419872, 1209627165996, 1801152661464

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Note that the sequence is not monotonically increasing, with a(4488) > a(4489) being the first of infinitely many examples. - Charles R Greathouse IV, Dec 28 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
T. H. Grönwall, Some asymptotic expressions in the Theory of Numbers, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013667, A013668, A013954-A013972, A017665-A017712.

[DivisorSigma(9,n): n in [1..20]]; // Bruno Berselli, Apr 10 2013

Table[DivisorSigma[9,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

PARI
```
a(n)=if(n<1,0,sigma(n,9))
```

[sigma(n,9)for n in range(1,21)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^9*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^8)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
n^9 + 1 <= a(n) < zeta(9)*n^9. In particular, Grönwall proves lim sup a(n)/n^9 = zeta(9) = A013667. - Charles R Greathouse IV, Dec 27 2021
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/264 = Bernoulli(10)/20. - Vaclav Kotesovec, May 07 2023
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s)*zeta(s-9).
Sum_{k=1..n} a(k) = zeta(10) * n^10 / 10 + O(n^11). (End)

A013956 a(n) = sigma_8(n), the sum of the 8th powers of the divisors of n.

Original entry on oeis.org

1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 25700456418, 37828630724, 55090232674, 78310985282, 110523825058, 152588281251

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013667, A013954-A013972, A017665-A017712.

[DivisorSigma(8,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

Table[DivisorSigma[8,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

a(n)=sigma(n,8) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,8)for n in range(1,21)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^8*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^7)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(8*e+8)-1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8).
Sum_{k=1..n} a(k) = zeta(9) * n^9 / 9 + O(n^10). (End)

A013961 a(n) = sigma_13(n), the sum of the 13th powers of the divisors of n.

Original entry on oeis.org

1, 8193, 1594324, 67117057, 1220703126, 13062296532, 96889010408, 549822930945, 2541867422653, 10001220711318, 34522712143932, 107006334784468, 302875106592254, 793811662272744, 1946196290656824, 4504149450301441, 9904578032905938, 20825519793796029, 42052983462257060

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_{p prime} ((p^((e(p)+1)*k)) - 1)/(p^k - 1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
By Fermat's little theorem n^13 == n (mod 13). Hence sigma_13(n) == sigma_(1) (mod 13). In fact, sigma_13(n) == sigma_(1) (mod 2730), where 2730 = 2*3*5*7*13 = the numerator of Bernoulli(12). - Peter Bala, Jan 12 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013672, A013954-A013972, A017665-A017712.

[DivisorSigma(13, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

A013961 := proc(n)
    numtheory[sigma][13](n) ;
end proc: # R. J. Mathar, Sep 21 2017

DivisorSigma[13, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^13*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 13); \\ Michel Marcus, Sep 10 2016

[sigma(n,13)for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^13*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-13)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24. - Simon Plouffe, Mar 01 2021
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(13*e+13)-1)/(p^13-1).
Sum_{k=1..n} a(k) = zeta(14) * n^14 / 14 + O(n^15). (End)

A013958 a(n) = sigma_10(n), the sum of the 10th powers of the divisors of n.

Original entry on oeis.org

1, 1025, 59050, 1049601, 9765626, 60526250, 282475250, 1074791425, 3486843451, 10009766650, 25937424602, 61978939050, 137858491850, 289537131250, 576660215300, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 10250010815226, 16680163512500, 26585860217050

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013669, A013954-A013972, A017665-A017712.

[DivisorSigma(10,n): n in [1..20]]; // Bruno Berselli, Apr 10 2013

lst={};Do[AppendTo[lst,DivisorSigma[10,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[10,Range[20]] (* Harvey P. Dale, Jan 04 2012 *)

a(n)=sigma(n,10) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,10)for n in range(1,18)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^10*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^9)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10)-1)/(p^10-1).
Dirichlet g.f.: zeta(s)*zeta(s-10).
Sum_{k=1..n} a(k) = zeta(11) * n^11 / 11 + O(n^12). (End)

A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

Original entry on oeis.org

1, 57, 703, 3641, 15501, 40071, 117307, 233017, 512461, 883557, 1770231, 2559623, 4824613, 6686499, 10897203, 14913081, 24132657, 29210277, 47039023, 56439141, 82466821, 100903167, 148023723, 163810951, 242203001, 275002941, 373584043

Offset: 1

Benoit Cloitre, Jun 21 2003

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001158, A013665, A013954.

Cf. A057660, A084218.

with(numtheory): a:=n->sigma[6](n^2)/sigma[3](n^2): seq(a(n),n=1..30); # Muniru A Asiru, Oct 09 2018

Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* Harvey P. Dale, May 02 2012 *)
f[p_, e_] := (p^(6*e + 3) + 1)/(p^3 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 13 2020 *)

a(n)=sumdiv(n^2,d,d^6)/sumdiv(n^2,d,d^3)

a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ Michel Marcus, Oct 09 2018

Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - Amiram Eldar, Sep 13 2020
Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - Vaclav Kotesovec, Sep 24 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = 90*zeta(7)/(7*Pi^4) = 0.133093... . - Amiram Eldar, Oct 30 2022
From Seiichi Manyama, May 18 2024: (Start)
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^3.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_6(d). (End)

A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

Original entry on oeis.org

1, 32769, 14348908, 1073774593, 30517578126, 470199366252, 4747561509944, 35185445863425, 205891146443557, 1000030517610894, 4177248169415652, 15407492847694444, 51185893014090758, 155572843119354936, 437893920912786408, 1152956690052710401, 2862423051509815794

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013674, A013954-A013972, A017665-A017712.

[DivisorSigma(15, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[15, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^15*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 15); \\ Amiram Eldar, Oct 29 2023

[sigma(n,15)for n in range(1,15)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^15*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-15)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(15*e+15)-1)/(p^15-1).
Sum_{k=1..n} a(k) = zeta(16) * n^16 / 16 + O(n^17). (End)

cp's OEIS Frontend

A013955 a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A050999 Sum of squares of odd divisors of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A013959 a(n) = sigma_11(n), the sum of the 11th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Sage

Formula

A013957 a(n) = sigma_9(n), the sum of the 9th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A013956 a(n) = sigma_8(n), the sum of the 8th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs