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

0,1

Comments

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

Links

Table of n, a(n) for n=0..105.

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.

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

Example

0.47168061361299786807523563308048208742592638200698...

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]]

Prog

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

Crossrefs

Cf. A000010, A001615, A013661, A065472, A162511, A173557.

Sequence in context: A016490 A244680 A200355 * A021216 A335006 A085508

Adjacent sequences: A307865 A307866 A307867 * A307869 A307870 A307871

Keyword

nonn,cons

Author

Amiram Eldar, May 02 2019

Extensions

More digits from Vaclav Kotesovec, Sep 19 2020

Status

approved