A013664 -id:A013664 - cp's OEIS frontend

A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			7.99601165442664514565252322930504700357640...

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

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666.

Cf. A347213, A347214, A347215, A347216, A347217, A347219, A347220.

RealDigits[Total[Zeta[Range[2, 8]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

Equals A347217 + A013666. - R. J. Mathar, May 27 2024

A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			8.9980200472527273600703759985374590640620...

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

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347220.

RealDigits[Total[Zeta[Range[2, 9]]], 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

Original entry on oeis.org

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

Offset: 1

Sean A. Irvine, Aug 24 2021

			9.9990146223805454454075219574377780810680...

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

Cf. A002117, A013661, A013662, A013663, A013664, A013665, A013666, A013667, A013668.

Cf. A347213, A347214, A347215, A347216, A347217, A347218, A347219.

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

A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

Original entry on oeis.org

1, 127, 1093, 8128, 19531, 138811, 137257, 520192, 796797, 2480437, 1948717, 8883904, 5229043, 17431639, 21347383, 33292288, 25646167, 101193219, 49659541, 158747968, 150021901, 247487059, 154764793, 568569856, 305171875, 664088461, 580865013, 1115624896

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - Álvar Ibeas, Oct 30 2015

Enrique Pérez Herrero and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Enrique Pérez Herrero)
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 7 of A263950.

Cf. A000010, A000203, A008683, A013664, A013665, A069092.

A160897 := proc(n)
    add(numtheory[mobius](n/d)*d^7,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160897(n)) ;
end do: # R. J. Mathar, Mar 14 2016

A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* Enrique Pérez Herrero, Oct 27 2010 *)
f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ Altug Alkan, Oct 30 2015

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(6*f[i,2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 27 2010
From Álvar Ibeas, Oct 30 2015: (Start)
Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
For squarefree n, a(n) = A000203(n^6). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .
Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 27 2010

A308637 Decimal expansion of Pi^3/Zeta(3).

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Aug 23 2019

Index entries for zeta function.

-----+---------------------------------
   n |  Zeta(n)
-----+---------------------------------
   2 |     Pi^2  / 6         = A013661.
   3 |     Pi^3  / 25.79...  = A002117.
   4 |     Pi^4  / 90        = A013662.
   5 |     Pi^5  / A309926   = A013663.
   6 |     Pi^6  / 945       = A013664.
   7 |     Pi^7  / A309927   = A013665.
   8 |     Pi^8  / 9450      = A013666.
   9 |     Pi^9  / A309928   = A013667.
  10 |     Pi^10 / 93555     = A013668.
  11 |     Pi^11 / A309929   = A013669.
  12 | 691*Pi^12 / 638512875 = A013670.
...
Cf. A002432, A091925, A276120 (Zeta(3)/Pi^3).

RealDigits[Pi^3/Zeta[3], 10, 100][[1]] (* Amiram Eldar, Aug 24 2019 *)

PARI
```
Pi^3/zeta(3)
```

Pi^3/Zeta(3) = A091925/A002117.

More terms from Amiram Eldar, Aug 24 2019

A343359 Decimal expansion of 1/zeta(6).

Original entry on oeis.org

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

Offset: 0

Karl-Heinz Hofmann, Apr 12 2021

Decimal expansion of 1/zeta(6), the inverse of A013664.
1/zeta(6) has a known closed-form formula (945/Pi^6) like 1/zeta(2) = 6/Pi^2 and 1/zeta(4) = 90/Pi^4.
1/zeta(6) is the probability that 6 randomly selected numbers will be coprime. - A.H.M. Smeets, Apr 13 2021

			0.982952592264580419804896564993924132951221515986...

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
Wikipedia, Riemann zeta function.
Index entries for transcendental numbers

Cf. A008683, A059956, A088453, A215267, A343308.

RealDigits[1/Zeta[6], 10, 100][[1]] (* Amiram Eldar, Apr 12 2021 *)

PARI
```
1/zeta(6) \\ A.H.M. Smeets, Apr 13 2021
```

Equals 1/A013664 = 945/Pi^6.
From Amiram Eldar, Jun 01 2023: (Start)
Equals Sum_{k>=1} mu(k)/k^6, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^6). (End)

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

Original entry on oeis.org

0, 1, 6, 22, 56, 133, 252, 484, 798, 1344, 2002, 3157, 4368, 6441, 8630, 12112, 15504, 21274, 26334, 35014, 42762, 55133, 65780, 84349, 98336, 123124, 143304, 176373, 201376, 247380, 278256, 336744, 379000, 451402, 502250, 600055, 658008, 775733, 855042

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013664, A032741, A065608, A069153, A363604, A363605.

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

a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+3,5).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A381653 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,2,2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 03 2025

			0.1907518241220842136964721118357975989

Index to constants which are multiple zeta values (2,2,2)

Cf. A381651, A381652.

Mathematica
```
RealDigits[3 Zeta[6]/16, 10, 105][[1]]
```
PARI
```
zetamult([2,2,2])
```

Equals 3*A013664/16.

A160895 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 7.

Original entry on oeis.org

1, 63, 364, 2016, 3906, 22932, 19608, 64512, 88452, 246078, 177156, 733824, 402234, 1235304, 1421784, 2064384, 1508598, 5572476, 2613660, 7874496, 7137312, 11160828, 6728904, 23482368, 12206250, 25340742, 21493836, 39529728, 21243690, 89572392

Offset: 1

N. J. A. Sloane, Nov 19 2009

a(n) is the number of lattices L in Z^6 such that the quotient group Z^6 / L is C_nm x (C_m)^5 (and also (C_nm)^5 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 6 of A263950.

Cf. A000010, A000203, A013663, A013664, A069091.

A160895 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(5*e-5)*(1+p+p^2+p^3+p^4+p^5) ; end do; a; end proc: # R. J. Mathar, Jul 10 2011

A160895[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(7-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 20 2010 *)
f[p_, e_] := p^(5*e - 5) * (p^6-1) / (p-1); ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 08 2022 *)

vector(50, n, sumdiv(n^5, d, if(ispower(d, 6), moebius(sqrtnint(d, 6))*sigma(n^5/d), 0))) \\ Altug Alkan, Oct 30 2014

a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(5*f[i,2]-5)*(1+p+p^2+p^3+p^4+p^5); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

a(n) = J_6(n)/J_1(n)=J_6(n)/phi(n)=A069091(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 20 2010
Multiplicative with a(p^e) = p^(5e-5)*(1+p+p^2+p^3+p^4+p^5). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^5). - Álvar Ibeas, Oct 30 2015
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^6, where c = (1/6) * Product_{p prime} (1 + (p^5-1)/((p-1)*p^6)) = 0.3203646372... .
Sum_{k>=1} 1/a(k) = zeta(5)*zeta(6) * Product_{p prime} (1 - 2/p^6 + 1/p^11) = 1.0195114923... . (End)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^6). - Ridouane Oudra, Apr 01 2025

Definition corrected by Enrique Pérez Herrero, Oct 20 2010

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

Original entry on oeis.org

32, 72, 108, 128, 200, 243, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1024, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3125, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4096, 4232, 4500, 4563, 4608

Offset: 1

V. Raman, Sep 07 2012

nonn

Powerful numbers (A001694) that are not squares of cubefree numbers (A004709), cubes of squarefree numbers (A062838), or 6th powers of primes (A030516). - Amiram Eldar, Feb 07 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A143610, A216426.

Cf. A001694, A002117, A004709, A013661, A013664, A062838, A085966.

With[{max = 5000}, Union[Table[i^2*j^3, {j, 2, max^(1/3)}, {i, 2, Sqrt[max/j^3]}] // Flatten]] (* Amiram Eldar, Feb 07 2023 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4, 3), for(a=2, sqrtint(lim\b^3), listput(v, a^2*b^3))); Set(v) \\ Charles R Greathouse IV, Jan 03 2014

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A216427(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j, b = isqrt(x), integer_nthroot(x,6)[0]
        l, c = 0, n+x-1+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(1, b+1))
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 1 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - P(6) = 0.12806919584708298724..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

cp's OEIS Frontend

A347218 Decimal expansion of Sum_{k=2..8} zeta(k).

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A347219 Decimal expansion of Sum_{k=2..9} zeta(k).

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A347220 Decimal expansion of Sum_{k=2..10} zeta(k).

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A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

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A308637 Decimal expansion of Pi^3/Zeta(3).

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A343359 Decimal expansion of 1/zeta(6).

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A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

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A381653 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,2,2).

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