A013664 -id:A013664 - cp's OEIS frontend

A101289 Inverse Moebius transform of 5-simplex numbers A000389.

1, 7, 22, 63, 127, 280, 463, 855, 1309, 2135, 3004, 4704, 6189, 9037, 11776, 16359, 20350, 27901, 33650, 44695, 53614, 68790, 80731, 103776, 118882, 148701, 171220, 210469, 237337, 292292, 324633, 393351, 438922, 522298, 576346, 690333, 749399

Offset: 1

Jonathan Vos Post, Mar 31 2006

From Georg Fischer, Aug 06 2025: (Start)
The general pattern is a(n) = Sum_{d|n} (Product_{k=0..m-1} d+k)/m! = Sum_{d|n} binomial(d+m-1, m) = Sum{d|n} Axxxxxx(d), with:
m  Axxxxxx  resulting sequence
------------------------------
2  A000217  A007437
3  A000292  A116963
4  A000332  A073570
5  A000389  A101289 (this sequence)
The other formulas generalize correspondingly.
A116989 uses A000579 and m=6 within a modified formula.
(End)

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

Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.

Cf. A000203, A001157, A001158, A001159, A001160.

a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ Seiichi Manyama, Apr 19 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021

a(n) = my(f = factor(n));  (sigma(f, 5) + 10*sigma(f, 4) + 35*sigma(f, 3) + 50*sigma(f, 2) + 24*sigma(f))/120; \\ Amiram Eldar, Dec 30 2024

a(n) = Sum_{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)/120 = Sum_{d|n} C(d+4,5) = Sum{d|n} A000389(d) = Sum_{d|n} (d^5+10*d^4+35*d^3+50*d^2+24*d)/120.
G.f.: Sum_{k>=1} x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k). - Seiichi Manyama, Apr 19 2021
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 10*sigma_4(n) + 35*sigma_3(n) + 50*sigma_2(n) + 24*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 10*zeta(s-4) + 35*zeta(s-3) + 50*zeta(s-2) + 24*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A284926 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^5.

Original entry on oeis.org

1, 31, 244, 991, 3126, 7564, 16808, 31711, 59293, 96906, 161052, 241804, 371294, 521048, 762744, 1014751, 1419858, 1838083, 2476100, 3097866, 4101152, 4992612, 6436344, 7737484, 9768751, 11510114, 14408200, 16656728, 20511150, 23645064, 28629152, 32472031, 39296688

Offset: 1

Seiichi Manyama, Apr 06 2017

Multiplicative because this sequence is the Dirichlet convolution of A000584 and A062157 which are both multiplicative. - Andrew Howroyd, Jul 20 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).
Index entries for sequences mentioned by Glaisher.

Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), this sequence (k=5), A284927 (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

Cf. A000584, A013664, A062157.

Table[Sum[(-1)^(n/d + 1)*d^5, {d, Divisors[n]}], {n, 50}] (* Indranil Ghosh, Apr 06 2017 *)
f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := (15*2^(5*e + 1) + 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *)

a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^5); \\ Indranil Ghosh, Apr 06 2017

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**5 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^k). - Ilya Gutkovskiy, Apr 07 2017
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (15*2^(5*e+1)+1)/31, and a(p^e) = (p^(5*e+5) - 1)/(p^5 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 31*zeta(6)/192 = 0.164258... . (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A056552 Powerfree kernel of cubefree part of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 1, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 3, 5, 26, 1, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 5, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 2, 55, 7, 57, 58, 59, 30, 61, 62, 21, 1, 65, 66, 67, 34, 69, 70, 71, 3, 73, 74, 15, 38, 77

Offset: 1

Henry Bottomley, Jun 25 2000

			a(32) = 2 because cubefree part of 32 is 4 and powerfree kernel of 4 is 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000189, A000578, A007947, A008834, A013664, A019555, A048798, A050985, A053149, A053150, A056551.

f[p_, e_] :=  p^If[Divisible[e, 3], 0, 1]; a[n_] := Times @@ (f @@@ FactorInteger[ n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, if (frac(f[k,2]/3), f[k,2] = 1, f[k,2] = 0)); factorback(f); \\ Michel Marcus, Feb 28 2019

a(n) = A007947(A050985(n)) = A019555(A050985(n)) = n/(A053150(n)*A000189(n)) = A019555(n)/A053150(n) = A056551(n)^(1/3).
If n = Product_{j} Pj^Ej then a(n) = Product_{j} Pj^Fj, where Fj = 0 if Ej is 0 or a multiple of 3 and Fj = 1 otherwise.
Multiplicative with a(p^e) = p^(if 3|e, then 0, else 1). - Mitch Harris, Apr 19 2005
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.3480772773... . - Amiram Eldar, Oct 28 2022
Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 16 2023

A157292 Decimal expansion of 315/(2*Pi^4).

Original entry on oeis.org

1, 6, 1, 6, 8, 9, 2, 2, 0, 5, 1, 1, 2, 7, 8, 2, 7, 9, 2, 2, 9, 1, 5, 6, 3, 3, 6, 4, 5, 7, 1, 1, 9, 4, 3, 2, 7, 3, 3, 7, 8, 7, 8, 7, 9, 1, 9, 4, 8, 0, 2, 6, 3, 7, 8, 1, 1, 1, 4, 6, 5, 5, 8, 6, 8, 3, 5, 8, 5, 1, 8, 7, 1, 3, 9, 9, 4, 2, 7, 4, 3, 9, 2, 2, 8, 9, 0, 0, 1, 5, 3, 9, 0, 0, 8, 2, 5, 2, 2, 6, 3, 6, 2, 7, 2

Offset: 1

R. J. Mathar, Feb 26 2009

Equals the asymptotic mean of the abundancy index of the 5-free numbers (numbers that are not divisible by a 5th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.61689220511... = (1+1/2^2+1/2^4)*(1+1/3^2+1/3^4)*(1+1/5^2+1/5^4)*(1+1/7^2+1/7^4)*...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Index entries for transcendental numbers.

Cf. A001248, A004709, A030514, A092744, A013661, A013664, A082020.

Maple
```
evalf(315/2/Pi^4) ;
```

RealDigits[N[Zeta[2]/Zeta[6], 150]][[1]] (* Geoffrey Critzer, Feb 16 2015 *)

315/2/Pi^4 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{p = primes} (1 + 1/p^2 + 1/p^4), whereas, the product over (1 + 2/p^2 + 1/p^4) equals A082020^2.
Equals A013661/A013664 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)).
Equals 315*A092744/2.
Equals Sum_{n>=1} 1/A004709(n)^2. - Geoffrey Critzer, Feb 16 2015

A344302 Number of cyclic subgroups of the group (C_n)^6, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 64, 365, 2080, 3907, 23360, 19609, 66592, 88817, 250048, 177157, 759200, 402235, 1254976, 1426055, 2130976, 1508599, 5684288, 2613661, 8126560, 7157285, 11338048, 6728905, 24306080, 12210157, 25743040, 21582653, 40786720, 21243691, 91267520, 29583457

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160895.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013664, A060648, A064969, A160895, A280184, A344219, A344303, A344304, A344305, A344306.

f[p_, e_] := 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160895(n) = sumdiv(n, d, moebius(n/d)*d^6)/eulerphi(n);
a(n) = sumdiv(n, d, a160895(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_6|n} phi(x_1)*phi(x_2)* ... *phi(x_6)/phi(lcm(x_1, x_2, ..., x_6)).
If p is prime, a(p) = 1 + (p^6 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} ((1-1/p^5)/(p^2*(1-1/p))) = 0.32592074105... . (End)

A347328 Decimal expansion of zeta(6) / zeta(3).

Original entry on oeis.org

8, 4, 6, 3, 3, 5, 1, 9, 3, 7, 0, 8, 6, 9, 4, 9, 0, 2, 9, 9, 4, 8, 9, 5, 7, 3, 6, 7, 3, 5, 9, 7, 9, 7, 8, 3, 2, 3, 6, 8, 0, 2, 9, 3, 8, 2, 7, 2, 2, 6, 3, 7, 1, 0, 3, 4, 2, 3, 4, 5, 3, 6, 2, 0, 8, 7, 4, 2, 8, 3, 1, 0, 3, 4, 0, 8, 5, 6, 9, 7, 7, 7, 6, 1, 0, 5, 9

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.84633519370869490299489573673597978323680...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A008836, A182448, A347329, A347330, A347331.

Cf. A002117, A013664.

RealDigits[Zeta[6] / Zeta[3], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals Sum_{k>=1} A008836(k) / k^3.

Equals Product_{p prime} 1/(1+p^(-3)). [corrected by Amiram Eldar, Jun 06 2023]

A275703 Decimal expansion of the Dirichlet eta function at 6.

Original entry on oeis.org

9, 8, 5, 5, 5, 1, 0, 9, 1, 2, 9, 7, 4, 3, 5, 1, 0, 4, 0, 9, 8, 4, 3, 9, 2, 4, 4, 4, 8, 4, 9, 5, 4, 2, 6, 1, 4, 0, 4, 8, 8, 5, 6, 9, 3, 4, 6, 9, 3, 2, 6, 8, 8, 8, 0, 3, 4, 8, 3, 3, 3, 9, 3, 2, 5, 4, 1, 9, 6, 8, 0, 2, 1, 8, 6, 2, 7, 1, 7, 1, 3, 5, 7, 3, 9, 3, 7, 2, 9, 1, 1, 2, 7, 9, 5, 5, 9, 4, 6, 4

Offset: 0

Terry D. Grant, Aug 05 2016

It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240. Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by Petros Hadjicostas, May 07 2020]

			31*(Pi^6)/30240 = 0.9855510912974...

Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).

Cf. A013664, A136677, A334605.

Mathematica
```
RealDigits[31*(Pi^6)/30240,10,100]
```

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016

eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.

eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020

A282751 Expansion of phi_{7, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 132, 2196, 16912, 78150, 289872, 823592, 2164800, 4802733, 10315800, 19487292, 37138752, 62748686, 108714144, 171617400, 277094656, 410338962, 633960756, 893872100, 1321672800, 1808608032, 2572322544, 3404825976, 4753900800, 6105469375, 8282826552

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), this sequence (phi_{7, 2}), A282753 (phi_{9, 2}).
Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A013974 (E_4*E_6 = E_10).
Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), this sequence (n^2*sigma_5(n)).
Cf. A013664.

Table[n^2 * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[Sum[k^7*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n^2*A001160(n) for n > 0.
a(n) = (2*A282101(n) - A282595(n) - A013974(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(6) * n^8 / 8. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^7*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

A347216 Decimal expansion of Sum_{k=2..6} zeta(k).

Original entry on oeis.org

5, 9, 8, 3, 5, 8, 5, 0, 2, 0, 8, 4, 6, 7, 7, 7, 9, 7, 9, 4, 3, 4, 0, 4, 0, 4, 4, 0, 9, 4, 6, 5, 9, 7, 7, 7, 8, 7, 1, 7, 5, 8, 5, 5, 5, 5, 0, 0, 0, 7, 9, 0, 6, 0, 1, 0, 2, 7, 4, 0, 7, 3, 4, 2, 0, 6, 4, 1, 0, 2, 0, 6, 5, 0, 6, 4, 7, 2, 2, 1, 9, 3, 0, 1, 9, 8, 1

Offset: 1

Sean A. Irvine, Aug 24 2021

			5.9835850208467779794340404409465977787175855550...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664.

Cf. A347213, A347214, A347215, A347217, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 6]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347217 Decimal expansion of Sum_{k=2..7} zeta(k).

Original entry on oeis.org

6, 9, 9, 1, 9, 3, 4, 2, 9, 8, 2, 2, 8, 7, 0, 0, 8, 0, 6, 2, 7, 3, 8, 3, 7, 9, 9, 0, 7, 9, 6, 3, 9, 4, 5, 3, 8, 3, 1, 7, 4, 4, 9, 1, 1, 5, 5, 6, 6, 0, 2, 9, 3, 0, 7, 4, 4, 4, 6, 9, 0, 4, 7, 8, 6, 3, 5, 7, 0, 3, 5, 4, 3, 3, 8, 2, 0, 7, 7, 9, 2, 8, 3, 6, 5, 9, 0

Offset: 1

Sean A. Irvine, Aug 24 2021

			6.9919342982287008062738379907963945383174491155660...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A002117, A013661, A013662, A013663, A013664, A013665.

Cf. A347213, A347214, A347215, A347216, A347218, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 7]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

cp's OEIS Frontend

A101289 Inverse Moebius transform of 5-simplex numbers A000389.

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A284926 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^5.

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A056552 Powerfree kernel of cubefree part of n.

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A157292 Decimal expansion of 315/(2*Pi^4).

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A344302 Number of cyclic subgroups of the group (C_n)^6, where C_n is the cyclic group of order n.

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A347328 Decimal expansion of zeta(6) / zeta(3).

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A275703 Decimal expansion of the Dirichlet eta function at 6.

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