A059376 -id:A059376 - cp's OEIS frontend

A069091 Jordan function J_6(n).

1, 63, 728, 4032, 15624, 45864, 117648, 258048, 530712, 984312, 1771560, 2935296, 4826808, 7411824, 11374272, 16515072, 24137568, 33434856, 47045880, 62995968, 85647744, 111608280, 148035888, 187858944, 244125000, 304088904, 386889048

Offset: 1

Benoit Cloitre, Apr 05 2002

Moebius transform of n^6. - Enrique Pérez Herrero, Sep 14 2010

a(n) is divisible by 504 = (2^3)*(3^3)*7 = A006863(3) except for n = 1, 2, 3 and 7. See Lugo. - Peter Bala, Jan 13 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069092 - A069095 (J_7 through J_10).
Cf. A065959.
Cf. A013665.

with(numtheory): seq(add(d^6 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]
A069091[n_IntegerQ]:=JordanTotient[n,6]; (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := p^(6*e) - p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^6*moebius(n/d)),","))

a(n) = Sum_{d|n} d^6*mu(n/d).
Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).
Dirichlet generating function: zeta(s-6)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A(x^n)/(1 - x^n)^7 = x + 63*x^2 + 728*x^3 + 4032*x^4 + 15624*x^5 + ..., where A(x) = x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6 is the 6th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

Original entry on oeis.org

1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741, 4776, 4929, 7101, 6877, 9546, 10121, 12069, 12189, 17284, 16145, 19881, 21321, 25826, 24417, 33669, 29821, 38224, 38889, 44361, 45021, 58386, 50689, 61893, 64061, 76884, 68961, 91089, 79549, 99234, 101781

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 3 of A343510.

Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

A343497:= func< n | (&+[d^3*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343497(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024

a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
A343497[n_]:= DivisorSum[n, #^3*EulerPhi[n/#] &]; Table[A343497[n], {n, 50}] (* G. C. Greubel, Jun 24 2024 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^3);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4))

def A343497(n): return sum(k^3*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343497(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^3.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 20 2024

A069095 Jordan function J_10(n).

Original entry on oeis.org

1, 1023, 59048, 1047552, 9765624, 60406104, 282475248, 1072693248, 3486725352, 9990233352, 25937424600, 61855850496, 137858491848, 288972178704, 576640565952, 1098437885952, 2015993900448, 3566920035096, 6131066257800

Offset: 1

Benoit Cloitre, Apr 05 2002

a(n) is divisible by 264 = (2^3)*3*11 = A006863(5), except for n = 1, 2, 3 or 11. See Lugo. - Peter Bala, Jan 13 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J-2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069091 - A069094 (J_6 through J_9).

Cf. A013669.

f:= n -> n^10*mul(1-1/p^10, p=numtheory:-factorset(n)):
map(f, [$1..30]); # Robert Israel, Jan 09 2015

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 10]; Array[f, 21]
f[p_, e_] := p^(10*e) - p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI
```
a(n) = sumdiv(n,d,d^10*moebius(n/d));
```

a(n) = Sum_{d|n} d^10*mu(n/d).
Multiplicative with a(p^e) = p^(10e)-p^(10(e-1)).
Dirichlet generating function: zeta(s-10)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^10*Product_{distinct primes p dividing n} (1-1/p^10). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^11 / (11*zeta(11)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^10 = 1/zeta(11).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^10/(p^10-1)^2) = 1.0009955309... (End)

A063453 Multiplicative with a(p^e) = 1 - p^3.

Original entry on oeis.org

1, -7, -26, -7, -124, 182, -342, -7, -26, 868, -1330, 182, -2196, 2394, 3224, -7, -4912, 182, -6858, 868, 8892, 9310, -12166, 182, -124, 15372, -26, 2394, -24388, -22568, -29790, -7, 34580, 34384, 42408, 182, -50652, 48006, 57096, 868, -68920, -62244, -79506, 9310, 3224, 85162, -103822, 182

Offset: 1

Vladeta Jovovic, Jul 26 2001

More generally, Dirichlet g.f. for Sum_{d|n} mu(d)*d^k, the Dirichlet inverse of the Jordan function J_k, is zeta(s)/zeta(s-k).
Apart from different signs also Sum_{d|n} core(d)^3*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002
Dirichlet inverse of A059376. - R. J. Mathar, Jul 15 2010

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A046970(r = 2), A189922 (r = 4).

Cf. A027748.

a063453 = product . map ((1 -) . (^ 3)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012

Jinvk := proc(n,k) local a,f,p ; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; a := a*(1-p^k) ; end do: a ; end proc:
A063453 := proc(n) Jinvk(n,3) ; end proc: # R. J. Mathar, Jul 04 2011
# second Maple program:
a:= n-> mul(1-i[1]^3, i=ifactors(n)[2]):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 26 2024

a[n_] := Total[MoebiusMu[#]*#^3& /@ Divisors[n]]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := (1-p^3); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

a(n) = sumdiv(n, d, moebius(d) * d^3); \\ Indranil Ghosh, Mar 11 2017

from math import prod
from sympy import primefactors
def A063453(n): return prod(1-p**3 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

a(n) = Sum_{d|n} mu(d)*d^3.
Dirichlet g.f.: zeta(s)/zeta(s-3).
A023900(n) | a(n). - R. J. Mathar, Mar 30 2011
a(n)= product_{p|n}(1-p^3), n>=2, p prime, a(1)=1.  a(n)= J_{-3}(n)*n^3, with the Jordan function J_k(n). See the Apostol reference, p. 48, exercise 17. - Wolfdieter Lang, Jun 16 2011.
G.f.: Sum_{k>=1} mu(k)*k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * sigma_2(d)^(-1) * sigma_1(n/d), where sigma_2(n)^(-1) = A053822(n) denotes the Dirichlet inverse of sigma_2(n). - Peter Bala, Jan 26 2024

A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Original entry on oeis.org

1, 8, 14, 36, 32, 112, 58, 148, 131, 256, 134, 504, 184, 464, 448, 596, 308, 1048, 382, 1152, 812, 1072, 554, 2072, 807, 1472, 1184, 2088, 872, 3584, 994, 2388, 1876, 2464, 1856, 4716, 1408, 3056, 2576, 4736, 1724, 6496, 1894, 4824, 4192, 4432, 2258, 8344

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 30 2001

Inverse Moebius transform of A160889. - Vladeta Jovovic, Nov 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 and arXiv, arXiv:1103.5861 [math.NT], 2011.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv:1203.6201 [math.GR], 2012.

Cf. A000010, A002117, A064803, A060648, A060724, A064767, A160889, A280184, A330595, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(3,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := Sum[EulerPhi[i] EulerPhi[j] (EulerPhi[k] / EulerPhi[LCM[i, j, k]]), {i, Divisors[n]}, {j, Divisors[n]}, {k, Divisors[n]}];
Array[a, 48] (* Jean-François Alcover, Dec 13 2018, after Vladeta Jovovic *)
f[p_, e_] := 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, eulerphi(i)*eulerphi(j)*eulerphi(k)/eulerphi(lcm(lcm(i, j), k))))); \\ Michel Marcus, Dec 14 2018

a160889(n) = sumdiv(n, d, moebius(n/d)*d^3)/eulerphi(n);
a(n) = sumdiv(n, d, a160889(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{i|n, j|n, k|n} phi(i)*phi(j)*phi(k)/phi(lcm(i, j, k)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = A002117 * A330595 / 3 = 0.700772... . (End)
a(n) = Sum_{d divides n} J_3(d)/phi(d) = Sum_{1 <= i, j, k <= n} 1/phi(n/gcd(i,j,k,n)), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 23 2024

Formula and more terms from Vladeta Jovovic, Oct 30 2001

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
Cf. A222171.

[0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-3).
Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

A328259 a(n) = n * sigma_2(n).

Original entry on oeis.org

1, 10, 30, 84, 130, 300, 350, 680, 819, 1300, 1342, 2520, 2210, 3500, 3900, 5456, 4930, 8190, 6878, 10920, 10500, 13420, 12190, 20400, 16275, 22100, 22140, 29400, 24418, 39000, 29822, 43680, 40260, 49300, 45500, 68796, 50690, 68780, 66300, 88400, 68962, 105000, 79550, 112728, 106470

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

Moebius transform of A027847.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A001157, A027847, A038040, A064987, A275585, A281372, A294362.

Table[n DivisorSigma[2, n], {n, 1, 45}]
nmax = 45; CoefficientList[Series[Sum[k^3 x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = n*sigma(n, 2); \\ Michel Marcus, Dec 02 2020

G.f.: Sum_{k>=1} k^3 * x^k / (1 - x^k)^2.
G.f.: Sum_{k>=1} k * x^k * (1 + 4 * x^k + x^(2*k)) / (1 - x^k)^4.
Dirichlet g.f.: zeta(s - 1) * zeta(s - 3).
Sum_{k=1..n} a(k) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Oct 09 2019
Multiplicative with a(p^e) = (p^(3*e+2) - p^e)/(p^2 - 1). - Amiram Eldar, Dec 02 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4 - (2*n^4 - 4*n^3 - 3*n^2 - n)*q^n - (8*n^3 - 4*n)*q^(2*n) + (2*n^4 + 4*n^3 - 3*n^2 + n)*q^(3*n) - n^4*q^(4*n) )/(1 - q^n)^4. Apply the operator x*d/dx twice, followed by the operator q*d/dq once, to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{k = 1..n} sigma_3( gcd(k, n) ) = Sum_{d divides n} sigma_3(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k <= n} sigma_1( gcd(i, j, k, n) ) = Sum_{d divides n} sigma_1(d) * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A351242 a(n) = n^3 * Sum_{p|n, p prime} 1/p^3.

Original entry on oeis.org

0, 1, 1, 8, 1, 35, 1, 64, 27, 133, 1, 280, 1, 351, 152, 512, 1, 945, 1, 1064, 370, 1339, 1, 2240, 125, 2205, 729, 2808, 1, 4591, 1, 4096, 1358, 4921, 468, 7560, 1, 6867, 2224, 8512, 1, 12221, 1, 10712, 4104, 12175, 1, 17920, 343, 16625, 4940, 17640, 1, 25515, 1456, 22464

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Dirichlet convolution of A010051(n) and n^3. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 35; a(6) = 6^3 * Sum_{p|6, p prime} 1/p^3 = 216 * (1/2^3 + 1/3^3) = 35.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), this sequence (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059376, A085964.

a[n_]:= n^3 * Sum[1/p^3,{p,Select[Divisors[n],PrimeQ]}]; Array[a,56] (* Stefano Spezia, Jul 14 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^3) \\ David A. Corneth, Jul 14 2025

a(A000040(n)) = 1.
G.f.: Sum_{k>=1} x^prime(k) * (1 + 4*x^prime(k) + x^(2*prime(k))) / (1 - x^prime(k))^4. - Ilya Gutkovskiy, Feb 05 2022
Dirichlet g.f. = zeta(s-3)*primezeta(s).  This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^3/n^s) Sum_{p|n} 1/p^3. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^3*(p*j)^(s-3)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-3) = zeta(s-3)*primezeta(s).  The result generalizes to higher powers of p in a(n). - Michael Shamos, Mar 01 2023
Sum_{k=1..n} a(k) ~ A085964 * n^4/4. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059376(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^3, where c = A010051.
a(p^k) = p^(3*k-3) for p prime and k>=1. (End)

A069093 Jordan function J_8(n).

Original entry on oeis.org

1, 255, 6560, 65280, 390624, 1672800, 5764800, 16711680, 43040160, 99609120, 214358880, 428236800, 815730720, 1470024000, 2562493440, 4278190080, 6975757440, 10975240800, 16983563040, 25499934720, 37817088000

Offset: 1

Benoit Cloitre, Apr 05 2002

a(n) is divisible by 480 = (2^5)*3*5 = A006863(4), except for n = 1, 2, 3 and 5. See Lugo. - Peter Bala, Jan 13 2024

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Lugo, A little number theory problem (2008)
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069091 - A069095 (J_6 through J_10)

Cf. A013667.

with(numtheory): seq(add(d^8 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 8]; Array[f, 25]
f[p_, e_] := p^(8*e) - p^(8*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(sumdiv(n,d,d^8*moebius(n/d)),","))

a(n) = Sum_{d|n} d^8*mu(n/d).
Multiplicative with a(p^e) = p^(8e)-p^(8(e-1)).
Dirichlet generating function: zeta(s-8)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^8*Product_{distinct primes p dividing n} (1-1/p^8). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ n^9 / (9*zeta(9)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^8 = 1/zeta(9).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^8/(p^8-1)^2) = 1.0040927606... (End)

A160889 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4.

Original entry on oeis.org

1, 7, 13, 28, 31, 91, 57, 112, 117, 217, 133, 364, 183, 399, 403, 448, 307, 819, 381, 868, 741, 931, 553, 1456, 775, 1281, 1053, 1596, 871, 2821, 993, 1792, 1729, 2149, 1767, 3276, 1407, 2667, 2379, 3472, 1723, 5187, 1893, 3724, 3627, 3871, 2257, 5824, 2793

Offset: 1

N. J. A. Sloane, Nov 19 2009

Dirichlet convolution of A000290 and the series of absolute values of A063441. - R. J. Mathar, Jun 20 2011

a(n) is the number of lattices L in Z^3 such that the quotient group Z^3 / L is C_nm x C_m x C_m (and also C_nm x C_nm x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

			There are 35 = A160870(4,3) lattices of volume 4 in Z^3. Among them, 28 give the quotient group C_4 and 7 give the quotient group C_2 x C_2. Hence, a(4) = 28 and a(2) = 7.
There are 2667 = A160870(32,3) lattices of volume 32 in Z^3. Among them, a(32) = 1792 give the quotient group C_32 (m=1); a(4) = 28 give C_8 x C_2 x C_2 (m=2); a(2) = 7 give C_4 x C_4 x C_2 (m=2).

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Index to Jordan function ratios J_k/J_1

Cf. A156304, A000203.

A160889[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(4-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Aug 22 2010 *)

vector(100, n, sumdiv(n^2, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n^2/d), 0))) \\ Altug Alkan, Oct 30 2015

Moebius transform of A064969. Multiplicative with a(p^e) = (p^2+p+1)*p^(2*e-2). - Vladeta Jovovic, Nov 21 2009
a(n) = J_3(n)/J_1(n)=J_3(n)/phi(n)=A059376(n)/A000010(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 22 2010
Dirichlet g.f.: zeta(s-2)*product_{primes p} (1+p^(1-s)+p^(-s)). - R. J. Mathar, Jun 20 2011
From Álvar Ibeas, Oct 30 2015: (Start)
a(n) = A254981(n^2). For squarefree n, a(n) = A000203(n^2).
a(n) = Sum_{d|n, n/d squarefree} d^2 * A000203(n/d).
(End)
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.748932997843245303033906997685114802259883493595480897273662144... - Vaclav Kotesovec, Dec 18 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p^2-1) * (p^2 + p + 1))) = 1.400940662893945919882073637564538872630336562726971915578687405304250550... - Vaclav Kotesovec, Sep 19 2020
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^3). - Ridouane Oudra, Mar 26 2025

Definition corrected by Vladeta Jovovic, Nov 21 2009

Typo in Mathematica program and formula fixed by Enrique Pérez Herrero, Oct 19 2010

cp's OEIS Frontend

A069091 Jordan function J_6(n).

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A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

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A069095 Jordan function J_10(n).

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A063453 Multiplicative with a(p^e) = 1 - p^3.

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A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

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A282097 Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.