A072691 -id:A072691 - cp's OEIS frontend

A197070 Decimal expansion of the Dirichlet eta-function at 3.

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

Offset: 0

R. J. Mathar, Oct 09 2011

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

			0.9015426773696957140498036211335874930737...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table II (4)
Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), series last equation.
Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
Wikipedia, Dirichlet eta function.

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Maple
```
3*Zeta(3)/4 ; evalf(%) ;
```

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

-polylog(3,-1) \\ Charles R Greathouse IV, Mar 28 2012

3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Equals 3*zeta(3)/4 = 3*A002117/4.
Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014
Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016
Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021
Equals -int_0^1 log(x)log(1+x)/x dx [Barbieri] - R. J. Mathar, Jun 07 2024

A033634 OddPowerSigma(n) = sum of odd power divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 43, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 43, 84, 144, 68, 54, 96, 144

Offset: 1

Yasutoshi Kohmoto

Odd power divisors of n are all the terms of A268335 (numbers whose prime power factorization contains only odd exponents) that divide n. - Antti Karttunen, Nov 23 2017

The Mobius transform is 1, 2, 3, 0, 5, 6, 7, 8, 0, 10, 11, 0, 13, 14, 15, 0, 17, 0, 19, 0, 21, 22, 23, 24, 0, 26, ..., where the places of zeros seem to be listed in A072587. - R. J. Mathar, Nov 27 2017

			The divisors of 7 are 1^1 and 7^1, which have only odd exponents (=1), so a(8) = 1+7 = 8. The divisors of 8 are 1^1, 2^1, 2^2 and 2^3; 2^2 has an even prime power and does not count, so a(8) = 1+2+8=11. The divisors of 12 are 1^1, 2^1, 3^1, 2^2, 2^1*3^1 and 2^2*3; 2^2 and 2^2*3 don't count because they have prime factors with even powers, so a(12) = 1+2+3+6 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A000203, A008864, A072587, A181150, A268335, A295316.
Cf. also A126849.
Cf. A065463, A072691.

A033634 := proc(n)
    a := 1 ;
    for d in ifactors(n)[2] do
        if type(op(2,d),'odd') then
            s := op(2,d) ;
        else
            s := op(2,d)-1 ;
         end if;
        p := op(1,d) ;
        a := a*(1+(p^(s+2)-p)/(p^2-1)) ;
    end do:
    a;
end proc: # R. J. Mathar, Nov 20 2010

f[e_] := If[OddQ[e], e+2, e+1]; fun[p_,e_] := 1 + (p^f[e] - p)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));
A033634(n) = sumdiv(n,d,A295316(d)*d); \\ Antti Karttunen, Nov 23 2017

Let n = Product p(i)^r(i) then a(n) = Product (1+[p(i)^(s(i)+2)-p(i)]/[p(i)^2-1]), where si=ri when ri is odd, si=ri-1 when ri is even. Special cases:
a(p) = 1+p for primes p, subsequence A008864.
a(p^2) = 1+p for primes p.
a(p^3) = 1+p+p^3 for primes p, subsequence A181150.
a(n) = Sum_{d|n} A295316(d)*d. - Antti Karttunen, Nov 23 2017
a(n) <= A000203(n). - R. J. Mathar, Nov 27 2017
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/(p*(p+1))) = A072691 * A065463 = 0.5793804... . - Amiram Eldar, Oct 27 2022

A222171 Decimal expansion of Pi^2/24.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 13 2013

			0.411233516712056609118103791661506297304737475301699609433889557342501867600...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press, 2006, p. 242.
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.45, p. 158 and 199-200.

Index entries for transcendental numbers

Cf. A000005, A013679, A111003, A072691, A078471, A369180.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*(pi)^2/24))); // Vincenzo Librandi, Sep 25 2015

Mathematica
```
RealDigits[Pi^2/24, 10, 100] // First
```
PARI
```
Pi^2/24 \\ Michel Marcus, Dec 10 2020
```

Equals Integral_{x=0..Pi/2} log(sec(x))/tan(x) dx.
Equals Sum_{k >= 1} 1/(2k)^2. - Geoffrey Critzer, Nov 02 2013
Equals (1/10) * Sum_{k>=1} d(k^2)/k^2, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, Jun 27 2020
Equals Sum_{n >= 0} 1/((2*n+1)*(6*n+3)). - Peter Bala, Feb 02 2022
Equals Sum_{n>=0} ((-1)^n * (Sum_{k>=n+1} (-1)^k/k)^2) (Furdui, 2013). - Amiram Eldar, Mar 26 2022
Equals Sum_{n>=1} A369180(n)/n^2. - Friedjof Tellkamp, Jan 23 2025

Leading 0 term removed (to make offset correct) by Rick L. Shepherd, Jan 01 2014

A173143 Partial sums of the squarefree numbers, A005117.

Original entry on oeis.org

1, 3, 6, 11, 17, 24, 34, 45, 58, 72, 87, 104, 123, 144, 166, 189, 215, 244, 274, 305, 338, 372, 407, 444, 482, 521, 562, 604, 647, 693, 740, 791, 844, 899, 956, 1014, 1073, 1134, 1196, 1261, 1327, 1394, 1463, 1533, 1604, 1677, 1751, 1828, 1906, 1985, 2067, 2150

Offset: 1

Jonathan Vos Post, Feb 10 2010

			The first squarefree numbers are: 1, 2, 3,  5,  6,  7, 10, ...
So, the first partial sums are:   1, 3, 6, 11, 17, 24, 34, ...

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

Cf. A005117, A072691.

Accumulate[Select[Range[100],SquareFreeQ]] (* Harvey P. Dale, Jan 09 2016 *)

lista(nn)=s = 0; for (n=1, nn, if (issquarefree(n), s += n; print1(s, ", "););); \\ Michel Marcus, Oct 01 2015

helper(n,k)=my(t=(n+1)\k); binomial(t,2)*k + (n+1 - t*k)*t
a(n)=my(s); forsquarefree(k=1,sqrtint(n), s+=moebius(k)*helper(n,k[1]^2)); s \\ Charles R Greathouse IV, Feb 05 2018

a(n) ~ (Pi^2/12) * n^2. - Amiram Eldar, Oct 21 2020

A267315 Decimal expansion of the Dirichlet eta function at 4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

			eta(4) = 1/1^4 - 1/2^4 + 1/3^4 - 1/4^4 + 1/5^4 - 1/6^4 + ... = 0.9470328294972459175765032344735219149279070829288860...

OEIS Wiki, Euler's alternating zeta function.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet Eta Function.

Cf. A002162, A013662, A072691, A120296, A197070, A334585.

pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // Vincenzo Librandi, Feb 04 2016

Mathematica
```
RealDigits[(7 Pi^4)/720, 10, 120][[1]]
```

7*Pi^4/720 \\ Michel Marcus, Feb 01 2016

s = RLF(0); s
RealField(110)(s)
for i in range(1,10000): s += -((-1)^i/((i)^4))
print(s) # Terry D. Grant, Aug 04 2016

eta(4) = Sum_{k > 0} (-1)^(k+1)/k^4 = (7*Pi^4)/720.

eta(4) = Lim_{n -> infinity} A120296(n)/A334585(n) = (7/8)*A013662. - Petros Hadjicostas, May 07 2020

A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24

Offset: 1

Jont Allen (jba(AT)research.att.com), May 25 2001

The union of N, 2N, 3N, ..., where N = {1, 2, 3, 4, 5, 6, ...}. In other words, the numbers {m*n, m >= 1, n >= 1} sorted into nondecreasing order.
Considering the maximal rectangle in each of the Ferrers graphs of partitions of n, a(n) is the smallest such maximal rectangle; a(n) is also an inverse of A006218. - Henry Bottomley, Mar 11 2002
The numbers in A003991 arranged in numerical order. - Matthew Vandermast, Feb 28 2003
Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - Michel Lagneau, Jan 04 2012
The number 1 appears only once, primes appear twice, squares of primes appear thrice. All other positive integers appear at least four times. - Alonso del Arte, Nov 24 2013

			Array begins:
   1
   2  2
   3  3
   4  4  4
   5  5
   6  6  6  6
   7  7
   8  8  8  8
   9  9  9
  10 10 10 10
  11 11
  12 12 12 12 12 12
  13 13
  14 14 14 14
  15 15 15 15
  16 16 16 16 16
  17 17
  18 18 18 18 18 18
  19 19
  20 20 20 20 20 20
  21 21 21 21
  22 22 22 22
  23 23
  24 24 24 24 24 24 24 24

N. J. A. Sloane, Table of n, a(n) for n = 1..7069
Hayato Kobayashi, Perplexity on Reduced Corpora, in: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, Baltimore, Maryland, USA, June 23-25 2014, Association for Computational Linguistics, 2014, pp. 797-806.

Cf. A000005. An inverse to A006218.

Cf. A027750, A056538, A072691.

with(numtheory); t1:=[]; for i from 1 to 1000 do for j from 1 to tau(i) do t1:=[op(t1),i]; od: od: t1:=sort(t1);

Flatten[Table[Table[n, {Length[Divisors[n]]}], {n, 30}]]

a(n)=if(n<0,0,t=1;while(sum(k=1,t,floor(t/k))Benoit Cloitre, Nov 08 2009

a(n) >= pi(n+1) for all n; a(n) >= pi(n) + 1 for all n >= 24 (cf. A098357, A088526, A006218, A052511). - N. J. A. Sloane, Oct 22 2008
a(n) = A027750(n) * A056538(n). - Charles Kusniec, Jan 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Jan 14 2024

More terms from Erich Friedman, Jun 01 2001

A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Dec 27 2004

From A.H.M. Smeets, Jun 12 2018: (Start)
The denominator of the k-th convergent obtained from a continued fraction of a constant, the terms of the continued fraction satisfying the Gauss-Kuzmin distribution, will tend to exp(k*A100199).
Similarly, the error between the k-th convergent obtained from a continued fraction of a constant, and the constant itself will tend to exp(-2*k*A100199). (End)
The term "Lévy's constant" is sometimes used to refer to this constant (Wikipedia). - Bernard Schott, Sep 01 2022

			1.1865691104156254528217229759472371205683565364720543359542542986528...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7, p. 54.

G. C. Greubel, Table of n, a(n) for n = 1..10000
R. M. Corless, Continued Fractions and Chaos, Amer. Math. Monthly 99, 203-215, 1992.
Eric Weisstein's World of Mathematics, Khinchin-Levy Constant.
Eric Weisstein's World of Mathematics, Lévy Constant.
Wikipedia, Lévy's constant.

Cf. A086702, A089729, A072691, A002162, A174606.

RealDigits[Pi^2/(12*Log[2]), 10, 100][[1]] (* G. C. Greubel, Mar 23 2017 *)

Pi^2/log(4096) \\ Charles R Greathouse IV, Aug 04 2016

Equals 1/A089729 = log(A086702) = A174606/2.
Equals ((Pi^2)/12)/log(2) = A072691 / A002162 = (Sum_{n>=1} ((-1)^(n+1))/n^2) / (Sum_{n>=1} ((-1)^(n+1))/n^1). - Terry D. Grant, Aug 03 2016
Equals (-1/log(2)) * Integral_{x=0..1} log(x)/(1+x) dx (from Corless, 1992). - Bernard Schott, Sep 01 2022

A245058 Decimal expansion of the real part of Li_2(I), negated.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 21 2014

This is the decimal expansion of the real part of the dilogarithm of the square root of -1. The imaginary part is Catalan's number (A006752).

5*Pi^2/24 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the even numbers. - Amiram Eldar, May 12 2023

			0.2056167583560283045590518958307531486523687376508498047169447786712509338004...

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..104 from Robert G. Wilson v)
R. J. Mathar, Some definite integrals over a power multiplied by four modified Bessel functions vixra:1606.0141 (2016) eq. (39)
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.3.13)

Cf. A006752, A007949, A072691, A222171, A242301.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/48; // G. C. Greubel, Aug 25 2018

RealDigits[ Re[ PolyLog[2, I]], 10, 111][[1]] (* or *) RealDigits[ Zeta[2]/8, 10, 111][[1]] (* or *) RealDigits[ Pi^2/48, 10, 111][[1]]

zeta(2)/8 \\ Charles R Greathouse IV, Aug 27 2014

(pi**2/48).n(200) # F. Chapoton, Mar 16 2020

Also equals -zeta(2)/8 = -Pi^2/48.
Also equals the Bessel moment Integral_{0..inf} x I_1(x) K_0(x)^2 K_1(x) dx. - Jean-François Alcover, Jun 05 2016
From Terry D. Grant, Sep 11 2016: (Start)
Equals Sum_{n>=0} (-1)^n/(2n+2)^2.
Equals (Sum_{n>=1} 1/(2n)^2)/2 = A222171/2. (End)
Equals Sum_{k>=1} A007949(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals a tenth of integral_0^{pi/2} arccos[cos x/(1+2 cos x)]dx [Nahin]. - R. J. Mathar, May 22 2024
Equals Integral_{x>=0} x/(exp(2*x) + 1) dx. - Kritsada Moomuang, May 29 2025

A214549 Decimal expansion of 4*Pi^2/27.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 20 2012

Represents the value of the Dirichlet series for A011655 (principal Dirichlet character mod 3) at s=2.

Equals the asymptotic mean of the abundancy index of the numbers that are not divisible by 3 (A001651). - Amiram Eldar, May 12 2023

			1.4621636149762012768643690370186...

G. C. Greubel, Table of n, a(n) for n = 1..10000
R. J. Mathar, Table of Dirichlet L-Series, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
Index entries for transcendental numbers.

Cf. A001651, A011655, A100044.

using Nemo
R = RealField(310)
t = const_pi(RR) + const_pi(RR); s = t * t
s / RR(27) |> println # Peter Luschny, Mar 13 2018

R:= RealField(); 4*Pi(R)^2/27; // G. C. Greubel, Mar 08 2018

R:=RealField(106); SetDefaultRealField(R); n:=4*Pi(R)^2/27; Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 13 2018

Maple
```
evalf(4*Pi^2/27) ;
```

RealDigits[(4Pi^2)/27,10,120][[1]] (* Harvey P. Dale, Dec 20 2012 *)

4*Pi^2/27 \\ G. C. Greubel, Mar 08 2018

Equals (4/3)*A100044.
Equals Sum_{n>=0} (1/(3*n+1)^2 + 1/(3*n+2)^2).
From Peter Luschny, May 13 2020: (Start)
Equals (8/9) * Sum_(k>=1) 1/k^2 =8/9 *A013661.
Equals -(16/9) * Sum_(k>=1) (-1)^k/k^2 = -16/9 * A072691.
Equals (64/27) * ( Integral_{x=0..1} sqrt(1 - x^2) )^2 = 64/27 * A091476. (End)
Equals Integral_{x=0..oo} log(x)/(x^3 - 1) dx. - Amiram Eldar, Aug 12 2020

A258947 Decimal expansion of the multiple zeta value (Euler sum) zetamult(6,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 15 2015

			0.01781974041683598836265953024872461216871313711029118841882136191761348...

Richard E. Crandall, Joe P. Buhler, On the evaluation of Euler Sums, Exp. Math. 3 (4) (1994) 275-285 Table 1.
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691, A197110.

digits = 99; zetamult[6,2] = NSum[HarmonicNumber[m-1, 2]/m^6, {m, 2, Infinity}, WorkingPrecision -> digits+20, NSumTerms -> 200, Method -> {"NIntegrate", "MaxRecursion" -> 18}]; Join[{0}, RealDigits[zetamult[6,2], 10, digits] // First]

zetamult([6,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult([2, 2, 1, 1, 1, 1]) \\ Charles R Greathouse IV, Feb 04 2025

zetamult(6,2) = Sum_{m>=2} (sum_{n=1..m-1} 1/(m^6*n^2)).

Equals Sum_{m>=2} H(m-1, 2)/m^6, where H(n,2) is the n-th harmonic number of order 2.

cp's OEIS Frontend

A197070 Decimal expansion of the Dirichlet eta-function at 3.

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A033634 OddPowerSigma(n) = sum of odd power divisors of n.

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A222171 Decimal expansion of Pi^2/24.

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A173143 Partial sums of the squarefree numbers, A005117.

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A267315 Decimal expansion of the Dirichlet eta function at 4.

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Magma

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A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.

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Maple

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