A367822 Decimal expansion of the asymptotic mean of psi(k)/phi(k), where psi(k) is the Dedekind psi function (A001615) and phi(k) is the Euler totient function (A000010).

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

Offset

1,1

Links

Table of n, a(n) for n=1..105.

V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, in: K. Alladi (ed.), Number Theory, Madras 1987, Springer, 1989, pp. 201-234, ResearchGate link.

Formula

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

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

Equals zeta(2) * Product_{p prime} (1 + 1/p^2 + 2/p^3).

Example

3.27957715098478360737291949891463398399913070810526...

Mathematica

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 2}, {0, 4, 6}, m]; RealDigits[2 * 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)))

Crossrefs

Cf. A000010, A001615, A013661, A307868 (mean of the inverse ratio).

Sequence in context: A026172 A026186 A026210 * A257326 A374611 A118966

Adjacent sequences: A367819 A367820 A367821 * A367823 A367824 A367825

Keyword

nonn,cons

Author

Amiram Eldar, Dec 02 2023

Status

approved