This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306633 #38 May 27 2024 15:45:44
%S A306633 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,
%T A306633 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,
%U A306633 4,4,9,7,9,7,8,7,6,0,1,9,4,2,2,8,2,2,0
%N A306633 Decimal expansion of zeta(2)/zeta(3).
%C A306633 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).
%C A306633 From _Amiram Eldar_, May 12 2023: (Start)
%C A306633 Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
%C A306633 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)
%H A306633 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
%H A306633 V. Sitaramaiah and M. V. Subbarao, <a href="https://doi.org/10.1007/BFb0086405">Some asymptotic formulae involving powers of arithmetic functions</a>, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, <a href="https://web.archive.org/web/20110910072314/http://www.math.ualberta.ca/~subbarao/documents/Subbarao3.pdf">alternative link</a>.
%F A306633 Equals A013661/A002117 = 1/A253905.
%F A306633 Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - _Amiram Eldar_, Jun 23 2020
%F A306633 Equals Product_{p prime} (1 + 1/(p*(p+1))). - _Amiram Eldar_, Aug 10 2020
%F A306633 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
%e A306633 1.3684327776202058757367658539847919411308139146524...
%t A306633 RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]
%o A306633 (PARI) zeta(2)/zeta(3) \\ _Michel Marcus_, Mar 04 2019
%Y A306633 Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.
%K A306633 nonn,cons
%O A306633 1,2
%A A306633 _Amiram Eldar_, Mar 02 2019