A013661 -id:A013661 - cp's OEIS frontend

A013679 Continued fraction for zeta(2) = Pi^2/6.

1, 1, 1, 1, 4, 2, 4, 7, 1, 4, 2, 3, 4, 10, 1, 2, 1, 1, 1, 15, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 3, 1, 1, 5, 1, 2, 2, 1, 1, 6, 27, 20, 3, 97, 105, 1, 1, 1, 1, 1, 45, 2, 8, 19, 1, 4, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 1, 2, 3, 6, 1, 1, 1, 2, 1, 5, 1, 1, 2, 9, 5, 3, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane

			1.644934066848226436472415166... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...))))

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

T. D. Noe, Table of n, a(n) for n = 0..9999
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for zeta function.

Cf. A013661 (decimal expansion).

Cf. continued fractions for zeta(3)-zeta(20): A013631, A013680-A013696.

Mathematica
```
ContinuedFraction[ Pi^2/6, 100]
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^2/6); for (n=1, 20000, write("b013679.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 29 2009

Offset changed by Andrew Howroyd, Jul 10 2024

A077641 Number of squarefree integers in closed interval [n, 2n-1], i.e., among n consecutive numbers beginning with n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 15, 16, 16, 17, 18, 19, 19, 19, 20, 21, 21, 22, 23, 23, 23, 24, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 37, 38, 38, 39, 38, 39, 40, 41, 41, 41, 42, 43, 43, 44, 45, 45

Offset: 1

Labos Elemer, Nov 14 2002

nonn

			For n = 10: among the numbers {10,...,19} seven are squarefree: {10,11,13,14,15,17,19}, so a(10) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A008683, A008966, A013661.

Table[Apply[Plus, Table[Abs[MoebiusMu[w+j]], {j, 0, w-1}]], {w, 1, 128}]
Table[Count[Range[n,2n-1],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 27 2013 *)
Module[{nn=80,sf},sf=Table[If[SquareFreeQ[n],1,0],{n,2nn}];Table[Total[ Take[ sf,{i,2i-1}]],{i,nn}]] (* Harvey P. Dale, May 20 2016 *)

a(n) = sum(i = 0, n-1, issquarefree(n+i)); \\ Amiram Eldar, Feb 25 2025

a(n) = Sum_{j=0..n-1} abs(mu(n+j)).
a(1) = 1; a(n + 1) = a(n) - issquarefree(n) + issquarefree(2n-2) + issquarefree(2n-1) for n > 0. - David A. Corneth, May 20 2016
a(n) ~ n/zeta(2). - Amiram Eldar, Feb 25 2025

A086463 Decimal expansion of Pi^2/18.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - Mats Granvik, Mar 19 2012

			0.548311355616075478824138388882008396406316633735...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.

J. M. Borwein and R. Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005) 25-36.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, arXiv:math/0206132 [math.PR], 2002.
Ji-Cai Liu, On two congruences involving Franel numbers, arXiv:2002.03650 [math.NT], 2020.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27.
Eric Weisstein's World of Mathematics, Bootstrap Percolation.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers.

Cf. A007814, A073010, A073016, A086464, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/18,10,120][[1]] (* Harvey P. Dale, Aug 14 2011 *)

Pi^2/18 \\ Charles R Greathouse IV, Mar 20 2012

Sum[1/n^2/Binomial[2n,n], {n,Infinity}].
Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - Mats Granvik, Mar 19 2012
Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - Michel Marcus, Feb 11 2020
Equals Sum_{k>=1} A007814(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - R. J. Mathar, Feb 07 2024
Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - Peter Bala, Feb 16 2024

A229099 Decimal expansion of 1 - 6/Pi^2.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Sep 13 2013

Probability that a random number is not squarefree; probability that two random numbers have a common divisor greater than 1.

			0.39207289814597337133672322074163416657384735196652070692634580863496...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A000796, A002388, A013661, A059956, A063035.

RealDigits[1-1/Zeta[2],10,90][[1]] (* Stefano Spezia, Feb 21 2025 *)

PARI
```
1-6/Pi^2
```

Equals 1 - 1/zeta(2). - Stefano Spezia, Feb 21 2025

A104141 Decimal expansion of 3/Pi^2.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, Mar 07 2005

3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the n-th Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy. - Alonso del Arte, May 28 2011
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378). - Amiram Eldar, May 22 2020

			3/Pi^2 = 0.303963550927013314331638389629...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156.
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.

Eric Weisstein's World of Mathematics, Farey Sequence
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
Index entries for transcendental numbers.

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.

l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)

3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013

Equals Sum_{n>=1} 1/A039956(n)^2. - Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (-1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2. (End)
From Hugo Pfoertner, Apr 23 2024: (Start)
Equals A059956/2.
Equals A082020/5. (End)

More terms from Ryan Propper, Aug 04 2005

A327527 Number of uniform divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

			The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

See link for additional cross-references.
Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.
Cf. A072774, A327526.
Cf. A001620, A013661, A306016, A368250, A368251.

Table[Length[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Dec 19 2023 *)

isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
A327527(n) = sumdiv(n,d,isA072774(d)); \\ Antti Karttunen, Nov 13 2021

From Amiram Eldar, Dec 19 2023: (Start)
a(n) = A034444(n) + A368251(n).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2) + c * zeta(2)), where gamma is Euler's constant (A001620) and c = A368250. (End)

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A000026 Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 6, 6, 10, 11, 12, 13, 14, 15, 8, 17, 12, 19, 20, 21, 22, 23, 18, 10, 26, 9, 28, 29, 30, 31, 10, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 44, 30, 46, 47, 24, 14, 20, 51, 52, 53, 18, 55, 42, 57, 58, 59, 60, 61, 62, 42, 12, 65, 66, 67, 68, 69, 70, 71, 36

Offset: 1

N. J. A. Sloane

a(n) = n if n is squarefree.

a(2n) = 2n if and only if n is squarefree. - Peter Munn, Feb 05 2017

			24 = 2^3*3^1, a(24) = 2*3*3*1 = 18.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. A. Gillman, The Average Size of a Certain Arithmetic Function, A6660 solution, Amer. Math. Monthly, 100 (1993), pp. 296-298.
B. Gordon and M. M. Robertson, Two theorems on mosaics, Canad. J. Math., 17 (1965), 1010-1014.
A. A. Mullin, Some related number-theoretic functions, Research Problem 4, Bull. Amer. Math. Soc., 69 (1963), 446-447.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Cf. A005117, A005361, A007947, A008474, A013661, A193551.

a000026 n = f a000040_list n 1 (0^(n-1)) 1 where
   f _  1 q e y  = y * e * q
   f ps'@(p:ps) x q e y
     | m == 0    = f ps' x' p (e+1) y
     | e > 0     = f ps x q 0 (y * e * q)
     | x < p * p = f ps' 1 x 1 y
     | otherwise = f ps x 1 0 y
     where (x', m) = divMod x p
a000026_list = map a000026 [1..]
-- Reinhard Zumkeller, Aug 27 2011

A000026 := proc(n) local e,j; e := ifactors(n)[2]:
mul(e[j][1]*e[j][2], j=1..nops(e)) end:
seq(A000026(n), n=1..80); # Peter Luschny, Jan 17 2011

Array[ Times@@Flatten[ FactorInteger[ # ] ]&, 100 ]

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],f[k,1]*f[k,2]))

a(n)=my(f=factor(n)); factorback(f[,1])*factorback(f[,2]) \\ Charles R Greathouse IV, Apr 04 2016

from math import prod
from sympy import factorint
def a(n): f = factorint(n); return prod(p*f[p] for p in f)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, May 27 2021

n = Product (p_j^k_j) -> a(n) = Product (p_j * k_j).
Multiplicative with a(p^e) = p*e. - David W. Wilson, Aug 01 2001
a(n) = A005361(n) * A007947(n). - Enrique Pérez Herrero, Jun 24 2010
a(A193551(n)) = n and a(m) != n for m < A193551(n). - Reinhard Zumkeller, Aug 27 2011
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)^2/2) * Product_{p prime} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.4175724194... . - Amiram Eldar, Oct 25 2022

Example, program, definition, comments and more terms added by Olivier Gérard (02/99).

A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 07 2006

Known to be transcendental. After n=2 it is the same as A100554(n).

Imaginary part of psi(I) (for the real part, see A248177). - Stanislav Sykora, Oct 03 2014

			2.076674047468581174134050794750000490445656266403816665575062484390...

Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Digamma function.
Index entries for transcendental numbers.

Cf. A013661 (Sum_{i>=1} 1/i^2), A232883 (Sum_{i>=0} 1/(2*i^2+1)). - Bruno Berselli, Dec 02 2013
Cf. A248177.
Essentially the same as A100554 and A259171.

RealDigits[N[Im[PolyGamma[0,I]],105]][[1]]  (* Vaclav Kotesovec, Oct 03 2014 *)

PARI
```
1/2+Pi/tanh(Pi)/2
```

imag(psi(I)) \\ Stanislav Sykora, Oct 03 2014

sumnumrat(1/(x^2+1),0) \\ Charles R Greathouse IV, Jan 20 2022

Equals 1/2 + Pi /(2*tanh(Pi)).
Equals 1+Integral_{x >= 0} sin(x)/(exp(x)-1) dx. - Robert FERREOL, Jan 12 2016.
Equals Sum_{k>=0} (-1)^(k+1)*(zeta(2*k) - 1). - Amiram Eldar, Apr 28 2025

Offset changed from 0 to 1 by Bruno Berselli, Dec 02 2013

A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...

There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.

			1.0561821217268161417379307653162198905...

S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A004613, A004614, A013661, A175646.

digits = 105;
LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/  DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];
RealDigits[1/(4*LandauRamanujanK/Pi)^2, 10, digits][[1]] (* Jean-François Alcover, Jan 12 2021 *)

Equals 1/A088539. - Vaclav Kotesovec, May 05 2020
From Amiram Eldar, Sep 27 2020: (Start)
Equals Sum_{k>=1} 1/A004613(k)^2.
The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)

More digits from Vaclav Kotesovec, Jun 27 2020

A306633 Decimal expansion of zeta(2)/zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 02 2019

Equals the asymptotic mean of the unitary abundancy index, lim_{n->oo} (1/n) * Sum{k=1..n} usigma(k)/k, where usigma(k) is the sum of the unitary divisors of k (A034448).
From Amiram Eldar, May 12 2023: (Start)
Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
In general, the asymptotic mean of the abundancy index of the k-free numbers (numbers that are not divisible by a k-th power other than 1) is zeta(2)/zeta(k+1) (Jakimczuk and Lalín, 2022). (End)

			1.3684327776202058757367658539847919411308139146524...

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).
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.

RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]

zeta(2)/zeta(3) \\ Michel Marcus, Mar 04 2019

Equals A013661/A002117 = 1/A253905.
Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - Amiram Eldar, Jun 23 2020
Equals Product_{p prime} (1 + 1/(p*(p+1))). - Amiram Eldar, Aug 10 2020
Equals Sum_{k>=1} mu(k)^2/(k*psi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Dedekind psi function values, A001615). - Amiram Eldar, Aug 18 2020

cp's OEIS Frontend

A013679 Continued fraction for zeta(2) = Pi^2/6.

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A077641 Number of squarefree integers in closed interval [n, 2n-1], i.e., among n consecutive numbers beginning with n.

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A086463 Decimal expansion of Pi^2/18.

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A229099 Decimal expansion of 1 - 6/Pi^2.

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A104141 Decimal expansion of 3/Pi^2.

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A327527 Number of uniform divisors of n.

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Extensions

A000026 Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j).

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