A307868 Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

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

Offset: 0

Author

Amiram Eldar, May 02 2019

Keywords

• nonn
• cons

Comments

Also, the asymptotic mean of A162511. - Amiram Eldar, Sep 18 2022

Examples

			0.47168061361299786807523563308048208742592638200698...
		

Links

• 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, A013661, A065472, A071974, A162511, A173557.

Programs

Mathematica

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 1, 2}, {0, -4, 6}, m]; RealDigits[(2/3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

PARI

prodeulerrat(1 - 2/(p*(p+1))) \\ Vaclav Kotesovec, Sep 19 2020

Formula

Equals lim_{m->oo} (1/m)*Sum_{k=1..m} phi(k)/psi(k).

Equals Product_{p prime} (1 - 2/(p * (p+1))).

Equals A065472 / zeta(2). - Amiram Eldar, Sep 18 2022

Extensions

More digits from Vaclav Kotesovec, Sep 19 2020