A013954 -id:A013954 - cp's OEIS frontend

A013969 a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.

1, 2097153, 10460353204, 4398048608257, 476837158203126, 21936961102828212, 558545864083284008, 9223376434903384065, 109418989141972712413, 1000000476837160300278, 7400249944258160101212, 46005141850728850805428, 247064529073450392704414, 1171356134499851307229224

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

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

[DivisorSigma(21,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[21,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

vector(50, n, sigma(n,21)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,21)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^21*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. - Vaclav Kotesovec, May 07 2023
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).
Dirichlet g.f.: zeta(s)*zeta(s-21).
Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)

A321810 Sum of 6th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, 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).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=6 of A285425.
Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013665, A013954.

f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)

apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023

A351311 Sum of the 6th powers of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 4097, 1, 1, 1, 16781313, 1, 531442, 1, 4097, 1, 1, 1, 4097, 244140626, 1, 531442, 4097, 1, 1, 1, 16781313, 1, 1, 1, 2177317874, 1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 16781313, 13841287202, 244140626, 1, 4097, 1, 531442, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^6 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

			a(16) = 16781313; a(16) = Sum_{d^2|16} (d^2)^6 = (1^2)^6 + (2^2)^6 + (4^2)^6 = 16781313.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), this sequence (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A013954.

f[p_, e_] := (p^(12*(1 + Floor[e/2])) - 1)/(p^12 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

a(n) = Sum_{d^2|n} (d^2)^6.
Multiplicative with a(p) = (p^(12*(1+floor(e/2))) - 1)/(p^12 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-12).
Sum_{k=1..n} a(k) ~ (zeta(13/2)/13) * n^(13/2). (End)
a(n) = Sum_{d|n} d^6 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
a(n) = Sum_{d|n} lambda(d)*d^6*sigma_6(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

A013960 a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n.

Original entry on oeis.org

1, 4097, 531442, 16781313, 244140626, 2177317874, 13841287202, 68736258049, 282430067923, 1000244144722, 3138428376722, 8918294543346, 23298085122482, 56707753666594, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162

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, A013671, A013954-A013972, A017665-A017712.

[DivisorSigma(12, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[12,Range[20]] (* Harvey P. Dale, Jan 28 2015 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^12*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 12); \\ Amiram Eldar, Oct 29 2023

[sigma(n,12) for n in range(1,17)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^12*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-12)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(12*e+12)-1)/(p^12-1).
Sum_{k=1..n} a(k) = zeta(13) * n^13 / 13 + O(n^14). (End)

A013965 a(n) = sigma_17(n), the sum of the 17th powers of the divisors of n.

Original entry on oeis.org

1, 131073, 129140164, 17180000257, 762939453126, 16926788715972, 232630513987208, 2251816993685505, 16677181828806733, 100000762939584198, 505447028499293772, 2218628050709022148, 8650415919381337934, 30491579359845314184, 98526126098761952664

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, A013676, A013954-A013972, A017665-A017712.

[DivisorSigma(17, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[17,Range[20]] (* Harvey P. Dale, May 30 2013 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^17*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 17); \\ Amiram Eldar, Oct 29 2023

[sigma(n,17)for n in range(1,14)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^17*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-17)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 43867/28728, also equals Bernoulli(18)/36. - Simon Plouffe, May 06 2023
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(17*e+17)-1)/(p^17-1).
Sum_{k=1..n} a(k) = zeta(18) * n^18 / 18 + O(n^19). (End)

A015764 Numbers n such that phi(n) | sigma_6(n).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 262, 750, 786, 8646, 56946, 222386, 626406, 667158, 737286, 1223123, 2446246, 2939046, 3669369, 6804006, 7338738, 27798250, 31684246, 41697375, 44970486, 53817126, 62128086, 76745867, 83394750, 95052738, 139991987, 153491734, 174684203

Offset: 1

Robert G. Wilson v

nonn

sigma_6(n) is the sum of the 6th powers of the divisors of n.

Cf. A000010 (phi(n)), A013954 (sigma_6(n)).

Cf. A020492, A015759, A015761, A015762, A015763, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

with(numtheory): A015764:=n->`if`(sigma[6](n) mod phi(n) = 0,n,NULL): seq(A015764(n), n=1..10^5); # Wesley Ivan Hurt, Mar 10 2015

Select[Range[10^5], Divisible[DivisorSigma[6, #], EulerPhi[#]] &] (* Amiram Eldar, Jan 20 2019 *)

More terms from Labos Elemer, May 03 2002

a(23)-a(34) from Amiram Eldar, Jan 20 2019

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

A301544 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_6(k)).

Original entry on oeis.org

1, 1, 66, 796, 7102, 70178, 702813, 6439533, 56938814, 495807251, 4218728690, 34991240657, 284295574638, 2269120791410, 17804772970005, 137455131596032, 1045354069608726, 7839809431539193, 58027706392726849, 424187792875896932, 3064539107659680502

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1769

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), this sequence (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A013954, A301550.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[6, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(8 * 2^(3/8) * Pi * (Zeta(7)/15)^(1/8) * n^(7/8)/7 - Pi*(5/Zeta(7))^(1/8) * n^(1/8) / (504 * 2^(3/8) * 3^(7/8)) + 45*Zeta(7) / (16*Pi^6)) * Zeta(7)^(1/16) / (2^(29/16) * 15^(1/16) * n^(9/16)).

G.f.: exp(Sum_{k>=1} sigma_7(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A013967 a(n) = sigma_19(n), the sum of the 19th powers of the divisors of n.

Original entry on oeis.org

1, 524289, 1162261468, 274878431233, 19073486328126, 609360902796252, 11398895185373144, 144115462954287105, 1350851718835253557, 10000019073486852414, 61159090448414546292, 319480609006403630044, 1461920290375446110678, 5976315357844100294616

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013678, A013954-A013972, A017665-A017712.

[DivisorSigma(19,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[19,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

vector(50, n, sigma(n,19)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,19)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^19*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(19*e+19)-1)/(p^19-1).
Dirichlet g.f.: zeta(s)*zeta(s-19).
Sum_{k=1..n} a(k) = zeta(20) * n^20 / 20 + O(n^21). (End)

A017668 Denominator of sum of -2nd powers of divisors of n.

Original entry on oeis.org

1, 4, 9, 16, 25, 18, 49, 64, 81, 10, 121, 24, 169, 98, 45, 256, 289, 324, 361, 200, 441, 242, 529, 288, 625, 338, 729, 56, 841, 9, 961, 1024, 1089, 578, 49, 432, 1369, 722, 1521, 160, 1681, 441, 1849, 968, 2025, 1058, 2209, 1152, 2401, 500, 2601, 1352, 2809

Offset: 1

N. J. A. Sloane

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

			1, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Colin Defant, On the Density of Ranges of Generalized Divisor Functions, arXiv:1506.05432 [math.NT], 2015.

Cf. A017667 (numerator).

[Denominator(DivisorSigma(2,n)/n^2): n in [1..50]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-2, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[2, n]/n^2], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

a(n) = denominator(sigma(n, -2)); \\ Michel Marcus, Aug 24 2018

vector(50, n, denominator(sigma(n, 2)/n^2)) \\ G. C. Greubel, Nov 08 2018

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

cp's OEIS Frontend

A013969 a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.

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A321810 Sum of 6th powers of odd divisors of n.

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A351311 Sum of the 6th powers of the square divisors of n.

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A013960 a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n.

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A013965 a(n) = sigma_17(n), the sum of the 17th powers of the divisors of n.

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A015764 Numbers n such that phi(n) | sigma_6(n).

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

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