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
Examples
3.27957715098478360737291949891463398399913070810526...
Links
- 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.
Programs
-
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]] -
PARI
prodeulerrat(1 + 2/(p*(p-1)))
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).