A059378 -id:A059378 - cp's OEIS frontend

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

0, 1, 1, 32, 1, 275, 1, 1024, 243, 3157, 1, 8800, 1, 16839, 3368, 32768, 1, 66825, 1, 101024, 17050, 161083, 1, 281600, 3125, 371325, 59049, 538848, 1, 867151, 1, 1048576, 161294, 1419889, 19932, 2138400, 1, 2476131, 371536, 3232768, 1, 4629701, 1, 5154656, 818424, 6436375, 1

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^5. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.

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), A351244 (k=4), this sequence (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059378, A085966.

Array[#^5*DivisorSum[#, 1/#^5 &, PrimeQ] &, 47] (* Stefano Spezia, Jul 15 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^5) \\ David A. Corneth, Jul 15 2025

a(A000040(n)) = 1.
Dirichlet g.f.: zeta(s-5)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^5/n^s) Sum_{p|n} 1/p^5. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^5*(p*j)^(s-5)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-5) = zeta(s-5)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 03 2023
Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059378(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)^5, where c = A010051.
a(p^k) = p^(5*k-5) for p prime and k>=1. (End)

A351300 a(n) = n^5 * Product_{p|n, p prime} (1 + 1/p^5).

Original entry on oeis.org

1, 33, 244, 1056, 3126, 8052, 16808, 33792, 59292, 103158, 161052, 257664, 371294, 554664, 762744, 1081344, 1419858, 1956636, 2476100, 3301056, 4101152, 5314716, 6436344, 8245248, 9768750, 12252702, 14407956, 17749248, 20511150, 25170552, 28629152, 34603008, 39296688

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Sum of the 5th powers of the divisor complements of the squarefree divisors of n.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A008683 (mu).
Cf. A069095, A059378.
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), A065960 (k=4), this sequence (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

f[p_, e_] := p^(5*e) + p^(5*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Feb 08 2022 *)

PARI
```
a(n)=sumdiv(n, d, moebius(n/d)^2*d^5);
```

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

a(n) = Sum_{d|n} d^5 * mu(n/d)^2.
a(n) = n^5 * Sum_{d|n} mu(d)^2 / d^5.
Multiplicative with a(p^e) = p^(5*e) + p^(5*e-5). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^6 * zeta(6) / (6 * zeta(12)) = 225225 * n^6 / (1382 * Pi^6).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/(p^10-1)) = 1.03592823428850098309076014982275428113698561633329794485946580153004... (End)
a(n) = J_10(n) / J_5(n) = A069095(n) / A059378(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022

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)

A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 5 of A343510.

Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); 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)^5);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))

def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^5.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024

A059384 a(n) = Product_{i=1..n} J_5(i).

Original entry on oeis.org

1, 31, 7502, 7441984, 23248758016, 174412182636032, 2931171141381153792, 93047096712003345973248, 5471727569246068763302821888, 529903984716066283313298482921472, 85341036738522474927606720674503065600, 20487310643596659421020979792003903940198400

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)^5 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..120
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, A059378, A059381, A059382, A059383, A175836.

JordanTotient[n_Integer, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &];  f[n_] := Times @@ (JordanTotient[#, 5] & /@ Range[n]); (* Enrique Pérez Herrero *)  Array[f, 11] (* Robert G. Wilson v, Oct 08 2011 *)

A160893 a(n) = Sum_{d|n} Möbius(n/d)*d^5/phi(n).

Original entry on oeis.org

1, 31, 121, 496, 781, 3751, 2801, 7936, 9801, 24211, 16105, 60016, 30941, 86831, 94501, 126976, 88741, 303831, 137561, 387376, 338921, 499255, 292561, 960256, 488125, 959171, 793881, 1389296, 732541, 2929531, 954305, 2031616, 1948705

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^5 such that the quotient group Z^5 / L is C_nm x (C_m)^4 (and also (C_nm)^4 x C_m), for every m>=1. - Álvar Ibeas, Oct 30 2015

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

Column 5 of A263950.

Cf. A000010, A000203, A013662, A013663, A059378.

A160893 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(4*e-4)*(1+p+p^2+p^3+p^4) ; end do; a; end proc: # R. J. Mathar, Jul 10 2011

A160893[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(6-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
f[p_, e_] := p^(4*e - 4)*(1 + p + p^2 + p^3 + p^4); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 08 2022 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^5/eulerphi(n)); \\ Michel Marcus, Feb 15 2015

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^5 - 1)*f[i, 1]^(4*f[i, 2] - 4)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022

a(n) = J_5(n)/J_1(n) = J_5(n)/phi(n) = A059378(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 19 2010
Multiplicative with a(p^e) = p^(4e-4)*(1 + p+ p^2 + p^3 + p^4). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^4). - Álvar Ibeas, Oct 30 2015
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^5, where c = (1/5) * Product_{p prime} (1 + (p^4-1)/((p-1)*p^5)) = 0.3799167034... .
Sum_{k>=1} 1/a(k) = zeta(4)*zeta(5) * Product_{p prime} (1 - 2/p^5 + 1/p^9) = 1.0449010968... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^5). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 19 2010

A239633 Triangle read by rows: T(n,k) = A059384(n)/(A059384(k)*A059384(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 31, 1, 1, 242, 242, 1, 1, 992, 7744, 992, 1, 1, 3124, 99968, 99968, 3124, 1, 1, 7502, 756008, 3099008, 756008, 7502, 1, 1, 16806, 4067052, 52501944, 52501944, 4067052, 16806, 1, 1, 31744, 17209344, 533489664, 1680062208, 533489664, 17209344, 31744

Offset: 0

Tom Edgar, Mar 22 2014

We assume that A059384(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the Jordan totient function J_5 given in A059378.
Another name might be the 5-totienomial coefficients.

			The first five terms in the fifth Jordan totient function are 1,31,242,992,3124 and so T(4,2) = 992*242*31*1/((31*1)*(31*1))=7744 and T(5,3) = 3124*992*242*31*1/((242*31*1)*(31*1))=99968.
The triangle begins
1
1 1
1 31   1
1 242  242   1
1 992  7744  992   1
1 3124 99968 99968 3124 1

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A059378, A059384, A238453, A238688, A238743, A238754.

q=100 #change q for more rows
P=[0]+[i^5*prod([1-1/p^5 for p in prime_divisors(i)]) for i in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A059384(n)/(A059384(k)* A059384(n-k)).
T(n,k) = prod_{i=1..n} A059378(i)/(prod_{i=1..k} A059378(i)*prod_{i=1..n-k} A059378(i)).
T(n,k) = A059378(n)/n*(k/A059378(k)*T(n-1,k-1)+(n-k)/A059378(n-k)*T(n-1,k)).

A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.

Original entry on oeis.org

0, 1, 6, 16, 56, 71, 252, 296, 651, 721, 2002, 1282, 4368, 3402, 5782, 6672, 15504, 7947, 26334, 15702, 28868, 28457, 65780, 30212, 85580, 63063, 103284, 81452, 201376, 66102, 278256, 174624, 255794, 228684, 383166, 206838, 658008, 391419, 576394, 413244, 1086008

Offset: 1

Wolfdieter Lang, Jun 22 2011

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k.
This is the m=4 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.
The members for m=0, 1, 2 and 3 are A000010, A023896, A127415, and A189918, respectively, where in the m=1 and 2 cases the offset for n=1 should be taken as 0 (not 1).

			a(6) = A000332(4) + A000292(8)= 1 + 70 = 71.
a(6) = (6/6!)*(6*3666*(1/3) + 5*137*2 - 182) = 71.
a(12) = A000332(4) + A000332(8) + A000332(10) + A000332(14) = 1 + 70 + 210 + 1001 = 1282.
a(12) = (12/6!)*(12*18258*(1/3) + 5*407*2 - 182) = 1282.

Cf. A189918, A023896, A127415, A189922.

a(n) = sum(k=0, n-1, if (gcd(n,k) == 1, binomial(k+3, 4))); \\ Michel Marcus, Feb 01 2016

a(n) = sum(A000332(k+3), 0<=k<=n-1, gcd(k,n)=1), n>=1.

a(n) = (n/6!)*(n*(6*n^3+45*n^2+110*n+90)*P(1,n) + 5*(2*n^2+9*n+11)*P(-1,n) - P(-3,n)), n>=2, with P(k,n):= J(k,n)/n^k, where J(k,n) is the Jordan function (see A000010, A007434, A059376 - A059378, A069091 - A069095).

More terms from Michel Marcus, Feb 01 2016

cp's OEIS Frontend

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

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PARI

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A351300 a(n) = n^5 * Product_{p|n, p prime} (1 + 1/p^5).

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A069093 Jordan function J_8(n).

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Maple

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A069092 Jordan function J_7(n).

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A069094 Jordan function J_9(n).

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A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

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Magma

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PARI

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A059384 a(n) = Product_{i=1..n} J_5(i).

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