A059377 -id:A059377 - cp's OEIS frontend

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400

Offset: 0

N. J. A. Sloane, Feb 04 2017

The q-expansion of the square of this expression is given in A281371.

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

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

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

[0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

with(gfun):
with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)/720,q,M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024

Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).

Original entry on oeis.org

1, 17, 82, 272, 626, 1394, 2402, 4352, 6642, 10642, 14642, 22304, 28562, 40834, 51332, 69632, 83522, 112914, 130322, 170272, 196964, 248914, 279842, 356864, 391250, 485554, 538002, 653344, 707282, 872644, 923522, 1114112, 1200644

Offset: 1

N. J. A. Sloane, Dec 08 2001

E. Pérez Herrero, Table of n, a(n) for n=1..10000
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
Wikipedia, Dedekind Psi function.

Cf. A000010, A007434, A059377, A069093.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), this sequence (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

A065960 := proc(n) n^4*mul(1+1/p^4,p=numtheory[factorset](n)) ; end proc:
seq(A065960(n),n=1..20) ; # R. J. Mathar, Jun 06 2011

a[n_] := n^4*DivisorSum[n, MoebiusMu[#]^2/#^4&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := p^(4*e) + p^(4*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(n^4*sumdiv(n,d,moebius(d)^2/d^4),","))

Multiplicative with a(p^e) = p^(4*e)+p^(4*e-4). - Vladeta Jovovic, Dec 09 2001
a(n) = n^4 * Sum_{d|n} mu(d)^2/d^4. - Benoit Cloitre, Apr 07 2002
a(n) = J_8(n)/J_4(n) = A069093(n)/A059377(n), where J_k is the k-th Jordan Totient Function. - Enrique Pérez Herrero, Aug 29 2010
Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(2*s). - R. J. Mathar, Jun 06 2011
From Vaclav Kotesovec, Sep 19 2020: (Start)
Sum_{k=1..n} a(k) ~ 18711*zeta(5)*n^5 / Pi^10.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/(p^8-1)) = 1.078178802583045599985995264729541574821218371712364313741065126120993131... (End)

A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

Original entry on oeis.org

1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207, 70216, 83537, 114393, 130339, 172346, 199781, 249067, 279863, 364204, 393145, 485741, 545067, 659518, 707309, 887519, 923551, 1123472, 1216033, 1420129, 1514003, 1843746, 1874197

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 4 of A343510.

Cf. A000010, A001158 (sigma_3(n)), A018804, A054611, A069097, A332517, A342423, A342433, A343497, A343499, A343514.

A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343498(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^4);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 3));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))

def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343498(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^4.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.
Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 18 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)

A351244 a(n) = n^4 * Sum_{p|n, p prime} 1/p^4.

Original entry on oeis.org

0, 1, 1, 16, 1, 97, 1, 256, 81, 641, 1, 1552, 1, 2417, 706, 4096, 1, 7857, 1, 10256, 2482, 14657, 1, 24832, 625, 28577, 6561, 38672, 1, 61921, 1, 65536, 14722, 83537, 3026, 125712, 1, 130337, 28642, 164096, 1, 234193, 1, 234512, 57186, 279857, 1, 397312, 2401, 400625, 83602

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^4. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 97; a(6) = 6^4 * Sum_{p|6, p prime} 1/p^4 = 1296 * (1/2^4 + 1/3^4) = 97.

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), A351242 (k=3), this sequence (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059377, A085965.

a[n_]:= n^4 * Sum[1/p^4, {p, Select[Divisors[n], PrimeQ]}]; Array[a, 51] (* Stefano Spezia, Jul 14 2025 *)

a(A000040(n)) = 1.
G.f.: Sum_{k>=1} x^prime(k) * (1 + 11*x^prime(k) + 11*x^(2*prime(k)) + x^(3*prime(k))) / (1 - x^prime(k))^5. - Ilya Gutkovskiy, Feb 05 2022
Dirichlet g.f.: zeta(s-4)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^4/n^s) Sum_{p|n} 1/p^4. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^4*(p*j)^(s-4)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-4) = zeta(s-4)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 02 2023
Sum_{k=1..n} a(k) ~ A085965 * n^5/5. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059377(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)^4, where c = A010051.
a(p^k) = p^(4*k-4) for p prime and k>=1. (End)

A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 16, 41, 136, 157, 656, 401, 1096, 1121, 2512, 1465, 5576, 2381, 6416, 6437, 8776, 5221, 17936, 7241, 21352, 16441, 23440, 12721, 44936, 19657, 38096, 30281, 54536, 25261, 102992, 30785, 70216, 60065, 83536, 62957, 152456, 52061, 115856, 97621, 172072, 70645, 263056, 81401, 199240, 175997, 203536, 106081, 359816, 137601, 314512

Offset: 1

Laszlo Toth, Dec 28 2016

Inverse Moebius transform of A160891. - Seiichi Manyama, May 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013662, A060648, A064969, A160891, A280162, 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(4,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
Array[a, 50] (* Jean-François Alcover, Sep 28 2018 *)
f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ Michel Marcus, Feb 26 2018

a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
a(n) = sumdiv(n, d, a160891(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024

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)

A069092 Jordan function J_7(n).

Original entry on oeis.org

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset: 1

Benoit Cloitre, Apr 05 2002

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

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]
Cf. A013666.

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(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^7*moebius(n/d)), ", "))

a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

A069094 Jordan function J_9(n).

Original entry on oeis.org

1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368

Offset: 1

Benoit Cloitre, Apr 05 2002

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

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Jordan's totient function.

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A013668.

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
f[p_, e_] := p^(9*e) - p^(9*(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^9*moebius(n/d)),","))

a(n) = Sum_{d|n} d^9*mu(n/d).
Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).
Dirichlet generating function: zeta(s-9)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252...  (End)

A059383 Product J_4(i), i=1..n.

Original entry on oeis.org

1, 15, 1200, 288000, 179712000, 215654400000, 517570560000000, 1987470950400000000, 12878811758592000000000, 120545678060421120000000000, 1764788726804565196800000000000, 33883943554647651778560000000000000, 967725427920736934795673600000000000000

Offset: 1

N. J. A. Sloane, Jan 28 2001

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j)^4 for 1 <= i,j <= n. - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 203, #17.

G. C. Greubel, Table of n, a(n) for n = 1..140
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Eric Weisstein's World of Mathematics, Le Paige's Theorem

Cf. A001088, A059377, A059381, A059382, A059383, A059384, A175836.

JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/#]&]/; (n>0)&&IntegerQ[n]; A059383[n_]:=Times@@(JordanTotient[#, 4]&/@Range[n]); (* Enrique Pérez Herrero, Aug 12 2011 *)

cp's OEIS Frontend

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A065960 a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).

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A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

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A069095 Jordan function J_10(n).

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A351244 a(n) = n^4 * Sum_{p|n, p prime} 1/p^4.

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A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

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A069093 Jordan function J_8(n).

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