A013956 -id:A013956 - cp's OEIS frontend

A386788 a(n) = n^4*sigma_8(n).

0, 1, 4112, 531522, 16843008, 244141250, 2185618464, 13841289602, 68988964864, 282472589763, 1003908820000, 3138428391362, 8952429298176, 23298085151042, 56915382843424, 129766445482500, 282578800148480, 582622237313282, 1161527289105456, 2213314919196482, 4112073026880000

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..7500 from Vincenzo Librandi)

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386787.

Cf. A013956, A386751, A386778, A386782.

[0] cat [n^4*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Table[n^4*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = if (n, n^4*sigma(n,8), 0); \\ Michel Marcus, Aug 03 2025

G.f.: Sum_{k>=1} k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13.
a(n) = n^4*A013956(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-12). - R. J. Mathar, Aug 03 2025

A321812 Sum of 8th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 6562, 1, 390626, 6562, 5764802, 1, 43053283, 390626, 214358882, 6562, 815730722, 5764802, 2563287812, 1, 6975757442, 43053283, 16983563042, 390626, 37828630724, 214358882, 78310985282, 6562, 152588281251, 815730722, 282472589764

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=8 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013667, A013956.

a[n_] := DivisorSum[n, #^8 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321812(n)=sigma(n>>valuation(n,2),8), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321812(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),8)) # Chai Wah Wu, Jul 16 2022

a(n) = A013956(A000265(n)) = sigma_8(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)^8*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/18 = 0.0556671... . (End)

A386778 a(n) = n^2*sigma_8(n).

Original entry on oeis.org

0, 1, 1028, 59058, 1052688, 9765650, 60711624, 282475298, 1077952576, 3487315923, 10039088200, 25937424722, 62169647904, 137858492018, 290384606344, 576739757700, 1103823438080, 2015993900738, 3584960768844, 6131066258162, 10280182567200, 16682426149284, 26663672614216

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A282097, A386745, A282099, A386747, A282751, A386777, A282753.

Cf. A013956, A386751.

[0] cat [n^2*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^2*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^10*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^10*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = n^2*A013956(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A386782 a(n) = n^3*sigma_8(n).

Original entry on oeis.org

0, 1, 2056, 177174, 4210752, 48828250, 364269744, 1977327086, 8623620608, 31385843307, 100390882000, 285311671942, 746035774848, 1792160396234, 4065384488816, 8651096365500, 17661175009280, 34271896312546, 64529293839192, 116490258905078, 205603651344000, 350330949134964

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A282211, A386746, A282213, A386748, A282781, A386780, A386781.

Cf. A013956, A386751, A386778.

[0] cat [n^3*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^3*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^11*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^11*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = n^3*A013956(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-11). - R. J. Mathar, Aug 03 2025

A068025 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=8.

Original entry on oeis.org

1, 511, 9841, 174251, 488281, 6017605, 6725601, 50955971, 72636421, 276964061, 235794769, 2234070293, 883708281, 3698977205, 5148057541, 13910980083, 7411742281, 46982039533, 17927094321, 99343345101, 69493620405

Offset: 1

Vladeta Jovovic, Feb 08 2002

nonn

Cf. A067692, A068020-A068024, A068026-A068027, A000203, A001157-A001160, A013954-A013956.

CIP8 = CycleIndexPolynomial[SymmetricGroup[8], Array[x, 8]]; a[n_] := CIP8 /. x[k_] -> DivisorSigma[k, n]; Array[a, 21] (* Jean-François Alcover, Nov 04 2016 *)

1/8!*(sigma[1](n)^8 + 28*sigma[1](n)^6*sigma[2](n) + 112*sigma[1](n)^5*sigma[3](n) + 210*sigma[1](n)^4*sigma[2](n)^2 + 420*sigma[1](n)^4*sigma[4](n) + 1120*sigma[1](n)^3*sigma[2](n)*sigma[3](n) + 420*sigma[1](n)^2*sigma[2](n)^3 + 1344*sigma[1](n)^3*sigma[5](n) + 2520*sigma[1](n)^2*sigma[2](n)*sigma[4](n) + 1120*sigma[1](n)^2*sigma[3](n)^2 + 1680*sigma[1](n)*sigma[2](n)^2*sigma[3](n) + 105*sigma[2](n)^4 + 3360*sigma[1](n)^2*sigma[6](n) + 4032*sigma[1](n)*sigma[2](n)*sigma[5](n) + 3360*sigma[1](n)*sigma[3](n)*sigma[4](n) + 1260*sigma[2](n)^2*sigma[4](n) + 1120*sigma[2](n)*sigma[3](n)^2 + 5760*sigma[7](n)*sigma[1](n) + 3360*sigma[2](n)*sigma[6](n) + 2688*sigma[3](n)*sigma[5](n) + 1260*sigma[4](n)^2 + 5040*sigma[8](n)).

A373039 a(n) = (A372966(n) - 1)/240.

Original entry on oeis.org

0, 1, 27, 257, 1625, 6508, 24010, 65793, 177174, 391626, 893101, 1665644, 3398759, 5786411, 10531652, 16843009, 29065308, 42698935, 70764303, 100231882, 155608837, 215237342, 326294606, 426404460, 634767250, 819100920, 1162438641, 1480961067, 2084357107, 2538128133

Offset: 1

Hugo Pfoertner, May 20 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001159, A013956.

f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 0; a[n_] := (Times @@ f @@@ FactorInteger[n] - 1) / 240; Array[a, 30] (* Amiram Eldar, Jan 08 2025 *)

a(n) = (sigma(n^2, 8)/sigma(n^2, 4)-1)/240

From Amiram Eldar, Jan 08 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-8)/zeta(s-4) - 1)/240.
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(2160*zeta(5)) = 0.000447372... . (End)

A055702 Numbers n such that n | Sigma_8(n) + Phi(n)^8.

Original entry on oeis.org

1, 2, 6, 86, 2033, 9617, 32052, 439369, 552012, 708292, 849660, 1869252, 2038140, 2083244, 2350089, 2569210, 2930460, 3875508, 4973090, 7248671, 13864156, 23500890, 25516264, 45711708, 57226685, 109512060, 112389732, 197121708, 240926532, 386807715, 395172531

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_8(n) is the sum of the 8th powers of the divisors of n (A013956).

Do[If[Mod[DivisorSigma[8, n]+EulerPhi[n]^8, n]==0, Print[n]], {n, 1, 10^5}]

isok(n) = !((sigma(n, 8) + eulerphi(n)^8) % n); \\ Michel Marcus, Mar 02 2014

More terms from Michel Marcus, Mar 02 2014

A158033 a(n) = sigma_(Fibonacci(n)) (n).

Original entry on oeis.org

1, 3, 10, 73, 3126, 1686434, 96889010408, 9223376434903384065, 278128389443693527934467475898331, 10000000000000000277555756156289135105943945819724042094

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

Cf. A000045, A000203, A001157, A001158, A001160, A013956, A013961, A013969.

a:= n-> numtheory[sigma][combinat[fibonacci](n)](n):
seq(a(n), n=1..10);  # Alois P. Heinz, Feb 10 2020

Table[DivisorSigma[Fibonacci[n],n],{n,10}] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = sigma(n, fibonacci(n)); \\ Michel Marcus, Feb 09 2020

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

Name edited by Michel Marcus, Feb 09 2020

cp's OEIS Frontend

A386788 a(n) = n^4*sigma_8(n).

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A321812 Sum of 8th powers of odd divisors of n.

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A386778 a(n) = n^2*sigma_8(n).

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Magma

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A386782 a(n) = n^3*sigma_8(n).

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A068025 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=8.

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A373039 a(n) = (A372966(n) - 1)/240.

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A055702 Numbers n such that n | Sigma_8(n) + Phi(n)^8.

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A158033 a(n) = sigma_(Fibonacci(n)) (n).

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