cp's OEIS Frontend

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).

%I A367822 #6 Dec 02 2023 07:04:00
%S A367822 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,
%T A367822 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,
%U A367822 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
%N 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).
%H A367822 V. Sitaramaiah and M. V. Subbarao, <a href="https://doi.org/10.1007/BFb0086405">Some asymptotic formulae involving powers of arithmetic functions</a>, in: K. Alladi (ed.), Number Theory, Madras 1987, Springer, 1989, pp. 201-234, <a href="https://www.researchgate.net/profile/V_Sitaramaiah/publication/227069035_Some_asymptotic_formulae_involving_powers_of_arithmetic_functions">ResearchGate link</a>.
%F A367822 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} psi(k)/phi(k).
%F A367822 Equals Product_{p prime} (1 + 2/(p*(p-1))).
%F A367822 Equals zeta(2) * Product_{p prime} (1 + 1/p^2 + 2/p^3).
%e A367822 3.27957715098478360737291949891463398399913070810526...
%t A367822 $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]]
%o A367822 (PARI) prodeulerrat(1 + 2/(p*(p-1)))
%Y A367822 Cf. A000010, A001615, A013661, A307868 (mean of the inverse ratio).
%K A367822 nonn,cons
%O A367822 1,1
%A A367822 _Amiram Eldar_, Dec 02 2023