A308043 Decimal expansion of the asymptotic mean of 2^omega(k)/d(k), where omega(k) is the number of distinct prime divisors of k (A001221) and d(k) is its number of divisors (A000005).

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

Offset: 0

Author

Amiram Eldar, May 10 2019

Keywords

• nonn
• cons

Comments

Also the asymptotic mean of the ratio between the number of unitary divisors and the number of divisors of the integers.

Examples

			0.81919096414899190818036566038137358272226885247971...
		

Crossrefs

Cf. A000005, A001221, A034444, A307870 (mean of the inverse ratio).

Programs

Mathematica

$MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(2*p*Log[p/(p - 1)] - 1); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[f[2] * Exp[ NSum[ Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

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