A027860 -id:A027860 - cp's OEIS frontend

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

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)

A281788 a(n) = (A013967(n) - A037945(n))/174611.

Original entry on oeis.org

0, 3, 6656, 1574235, 109234176, 3489819540, 65281655808, 825351571995, 7736349470720, 57270269768634, 350259092774400, 1829670576438068, 8372440970643456, 34226453991167880, 126958657929489408, 432721923827171355, 1369171676955783168, 4056082931864408991, 11330441127202890240, 30026115193307387658, 75874353000273633280, 183636989491548765276

Offset: 1

Seiichi Manyama, Feb 03 2017

nonn

			a(1) = (1 - 1)/174611 = 0.
a(2) = (524289 - 456)/174611 = 3.
a(3) = (1162261468 - 50652)/174611 = 6656.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013967, A027860, A037945, A281876, A281928, A281956.

A281876 a(n) = (A013963(n) - A027364(n))/3617.

Original entry on oeis.org

0, 9, 3968, 296865, 8437248, 129997260, 1312568064, 9727799265, 56923182080, 276480648702, 1154893046400, 4259743681004, 14151477247488, 43011568291320, 121065502097664, 318760489739745, 791380439553024, 1865315725321293, 4197159808767360, 9059718006875214

Offset: 1

Seiichi Manyama, Feb 01 2017

nonn

			a(1) = (1 - 1)/3617 = 0.
a(2) = (32769 - 216)/3617 = 9.
a(3) = (14348908 - (-3348))/3617 = 3968.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013963, A027860, A027364, A281788, A281928, A281956.

A281928 a(n) = (A013965(n) - A037944(n))/43867.

Original entry on oeis.org

0, 3, 2944, 391635, 17392128, 385866060, 5303086848, 51332824275, 380176030720, 2279635315794, 11522261136000, 50576242992268, 197196432781824, 695091512105880, 2246019242126592, 6728295917456595, 18857917384178688, 49830812542200039

Offset: 1

Seiichi Manyama, Feb 02 2017

nonn

			a(1) = (1 - 1)/43867 = 0.
a(2) = (131073 - (-528))/43867 = 3.
a(3) = (129140164 - (-4284))/43867 = 2944.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013965, A027860, A037944, A281788, A281876, A281956.

A281956 a(n) = (A013969(n) - A037946(n))/77683.

Original entry on oeis.org

0, 27, 134656, 56615355, 6138243072, 282390755580, 7190065585152, 118730950577595, 1408531971420160, 12872835457479666, 95262154452748800, 592216338844654972, 3180419513581234176, 15078667591360144440, 64208193499209765888, 248996850497620053435

Offset: 1

Seiichi Manyama, Feb 03 2017

nonn

			a(1) = (1 - 1)/77683 = 0.
a(2) = (2097153 - (-288))/77683 = 27.
a(3) = (10460353204 - (-128844))/77683 = 134656.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013969, A027860, A037946, A281788, A281876, A281928.

A279889 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_5(n-k).

Original entry on oeis.org

0, 1, 66, 1577, 18218, 135550, 738236, 3207785, 11714718, 37347144, 106499470, 277489886, 668981686, 1512360404, 3228797252, 6570019945, 12793050456, 24001960051, 43483452090, 76485144056, 130752372320, 218220937122, 355664809556, 568293832670, 889969136158

Offset: 1

Seiichi Manyama, Dec 22 2016

nonn

In 1916, Ramanujan found the following identity. tau(n) = sigma_11(n) - 691/756 * (sigma_11(n) - sigma_5(n) + 252 * a(n)). This implies tau(n) == sigma_11(n) mod 691.

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory,Springer-Verlag, 1976. See p. 140, exercise 10.
Srinivasa Ramanujan, Collected papers, ed. G. H. Hardy et al., Cambridge, 1927, pp. 136-162.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
William Duke, Ramanujan and modular forms, in: K. Alladi et al., Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, Springer Cham, 2024. See eq. (6).
D. H. Lehmer, Some functions of Ramanujan, Math. Student, Vol. 27 (1959), pp. 105-116; entire volume. See p. 111, eq. (9).
Srinivasa Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.

Cf. Sum_{k=1..n-1} sigma_m(k)*sigma_m(n-k): A087115 (m=3), this sequence (m=5).

Cf. A000594, A001160, A013959, A027860.

a[n_] := (65 * DivisorSigma[11, n] + 691 * DivisorSigma[5, n] - 756 * RamanujanTau[n]) / 174132; Array[a, 25] (* Amiram Eldar, Jan 07 2025 *)

a(n) = sum(k=1, n-1, sigma(k, 5)*sigma(n-k, 5)) \\ Felix Fröhlich, Jan 01 2017

a(n) = {my(f = factor(n)); (65 * sigma(f, 11) + 691 * sigma(f, 5) - 756 * ramanujantau(n)) / 174132;} \\ Amiram Eldar, Jan 07 2025

A027860(n) = (sigma_11(n) - sigma_5(n) + 252*a(n))/756.

A281979 a(n) = (A281959(n) - A037947(n))/657931.

Original entry on oeis.org

0, 51, 1287808, 1711273635, 452970333696, 43211657266860, 2038311950075136, 57420813107839395, 1091144797392901120, 15199162675148592018, 164678453263146595200, 1449942615368630353516, 10725152052216567264768, 68394401763888606334680

Offset: 1

Seiichi Manyama, Feb 04 2017

nonn

			a(1) = (1 - 1)/657931 = 0.
a(2) = (33554433 - (-48))/657931 = 51.
a(3) = (847288609444 - (-195804))/657931 = 1287808.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A027860, A037947, A281788, A281876, A281928, A281956, A281959.

A337032 a(n) = (n*sigma_9(n) - tau(n))/7 = (A282254(n) - A000594(n))/7, where tau is Ramanujan's tau, sigma_9(n) = Sum_{d divides n} d^9.

Original entry on oeis.org

0, 150, 8400, 150300, 1394400, 8656200, 40356000, 153679800, 498153600, 1431378900, 3705270000, 8863150800, 19694152800, 41402744400, 82382680800, 157380332400, 288000115200, 511088547150, 875865085200, 1465721632200, 2382961862400, 3801687211800, 5918070367200, 9075809181600

Offset: 1

Jianing Song, Aug 12 2020

nonn

D. H. Lehmer shows that tau(n) == n*sigma_9(n) (mod 7), so a(n) is an integer for all n. Furthermore, if n == 3, 5, 6 (mod 7) then tau(n) == n*sigma_9(n) (mod 49). See the Wikipedia link below. It seems that the latter congruence also holds for most of the other numbers. Among the 571 numbers in [1, 1000] congruent to 0, 1, 2, 4 modulo 7, tau(n) == n*sigma_9(n) holds for 311 n's, and among the 5715 numbers in [1, 10000] congruent to 0, 1, 2, 4 modulo 7, the congruence holds for 3492 n's.
It seems that 150 divides a(n) for all n. There are no counterexamples for n <= 10000.
Number of n's in [2, N] which satisfy the higher-order congruence tau(n) == n*sigma_9(n) (mod 7^e) but not tau(n) == n*sigma_9(n) (mod 7^(e+1)):
  N = 1000:
   e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
  ---+----------------------+-------------------------+-------
   1 |                    0 |                     260 |   260
  ---+----------------------+-------------------------+-------
   2 |                  358 |                      80 |   438
  ---+----------------------+-------------------------+-------
   3 |                   45 |                     195 |   240
  ---+----------------------+-------------------------+-------
   4 |                   24 |                      28 |    52
  ---+----------------------+-------------------------+-------
   5 |                    2 |                       5 |     7
  ---+----------------------+-------------------------+-------
   6 |                    0 |                      2* |     2
  * n = 686, 942.
  N = 10000:
   e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
  ---+----------------------+-------------------------+-------
   1 |                    0 |                    2223 |  2223
  ---+----------------------+-------------------------+-------
   2 |                 3368 |                     728 |  4096
  ---+----------------------+-------------------------+-------
   3 |                  466 |                    2280 |  2746
  ---+----------------------+-------------------------+-------
   4 |                  397 |                     384 |   781
  ---+----------------------+-------------------------+-------
   5 |                   46 |                      87 |   133
  ---+----------------------+-------------------------+-------
   6 |                    6 |                      12 |    18
  ---+----------------------+-------------------------+-------
   7 |                  2** |                       0 |     2
  ** n = 5185, 9021.

			a(2) = (n*sigma_9(2) - tau(2))/7 = (2*(1^9+2^9) - (-24))/7 = 1050/7 = 150;
a(3) = (n*sigma_9(3) - tau(3))/7 = (3*(1^9+3^9) - 252)/7 = 58800/7 = 8400.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Congruences for the tau function.

Cf. A000594, A282254, A027860.

a[n_] := (n * DivisorSigma[9, n] - RamanujanTau[n]) / 7; Array[a, 24] (* Amiram Eldar, Jan 10 2025 *)

a(n) = (n*sigma(n, 9) - polcoeff( x * eta(x + x * O(x^n))^24, n))/7;

cp's OEIS Frontend

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

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A281788 a(n) = (A013967(n) - A037945(n))/174611.

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A281876 a(n) = (A013963(n) - A027364(n))/3617.

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A281928 a(n) = (A013965(n) - A037944(n))/43867.

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A281956 a(n) = (A013969(n) - A037946(n))/77683.

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A279889 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_5(n-k).

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A281979 a(n) = (A281959(n) - A037947(n))/657931.

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A337032 a(n) = (n*sigma_9(n) - tau(n))/7 = (A282254(n) - A000594(n))/7, where tau is Ramanujan's tau, sigma_9(n) = Sum_{d divides n} d^9.

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