A013972 -id:A013972 - cp's OEIS frontend

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

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)

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

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

A017667 Numerator of sum of -2nd powers of divisors of n.

Original entry on oeis.org

1, 5, 10, 21, 26, 25, 50, 85, 91, 13, 122, 35, 170, 125, 52, 341, 290, 455, 362, 273, 500, 305, 530, 425, 651, 425, 820, 75, 842, 13, 962, 1365, 1220, 725, 52, 637, 1370, 905, 1700, 221, 1682, 625, 1850, 1281, 2366, 1325, 2210, 1705, 2451, 651, 2900, 1785

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
Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
C. Defant proves that there are no positive integers n such that sigma_{-2}(n) lies in (Pi^2/8, 5/4). See arxiv link. - Michel Marcus, Aug 24 2018

			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, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 3 (2015), pp. 80-87; arXiv preprint, arXiv:1506.05432 [math.NT], 2015.

Cf. A017668 (denominator), A002117, A013661, A111003 (Pi^2/8).

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

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

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

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

Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - Franklin T. Adams-Watters, Sep 11 2005
sup_{n>=1} a(n)/A017668(n) = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - Amiram Eldar, Apr 02 2024

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

Original entry on oeis.org

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

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

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A013962 a(n) = sigma_14(n), the sum of the 14th powers of the divisors of n.

Original entry on oeis.org

1, 16385, 4782970, 268451841, 6103515626, 78368963450, 678223072850, 4398314962945, 22876797237931, 100006103532010, 379749833583242, 1283997101947770, 3937376385699290, 11112685048647250, 29192932133689220, 72061992352890881, 168377826559400930

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, A013673, A013954-A013972, A017665-A017712.

[DivisorSigma(14, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[14,Range[20]] (* Harvey P. Dale, Mar 10 2013 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^14*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 14); \\ Amiram Eldar, Oct 29 2023

[sigma(n,14) for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^14*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-14)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(14*e+14)-1)/(p^14-1).
Sum_{k=1..n} a(k) = zeta(15) * n^15 / 15 + O(n^16). (End)

A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

Original entry on oeis.org

1, 262145, 387420490, 68719738881, 3814697265626, 101560344351050, 1628413597910450, 18014467229220865, 150094635684419611, 1000003814697527770, 5559917313492231482, 26623434909949071690, 112455406951957393130, 426880482624234915250, 1477891883850485076740

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, A013677, A013954-A013972, A017665-A017712.

[DivisorSigma(18,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[18,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

a(n)=sigma(n,18) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,18)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^18*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(18*e+18)-1)/(p^18-1).
Dirichlet g.f.: zeta(s)*zeta(s-18).
Sum_{k=1..n} a(k) = zeta(19) * n^19 / 19. + O(n^20). (End)

A013971 a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.

Original entry on oeis.org

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 789730317205170252, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 100000011920928963466734, 895430243255237372246532, 6624738056749922960468044, 41753905413413116367045798

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 (first 1000 terms from Harvey P. Dale)
Index entries for sequences related to sigma(n).

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

[DivisorSigma(23,n): n in [1..30]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[23,Range[15]] (* Harvey P. Dale, May 02 2016 *)

vector(30, n, sigma(n,23)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,23)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^23*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).
Dirichlet g.f.: zeta(s)*zeta(s-23).
Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)

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

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PARI

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

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Sage

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

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Magma

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PARI

PARI

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A017667 Numerator of sum of -2nd powers of divisors of n.

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