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

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

Author

Amiram Eldar, Mar 02 2019

Keywords

• nonn
• cons

Comments

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)

Examples

			1.3684327776202058757367658539847919411308139146524...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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