A308043 - OEIS

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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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

Table of n, a(n) for n=0..86.

FORMULA

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

EXAMPLE

0.81919096414899190818036566038137358272226885247971...

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]]

CROSSREFS

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

Sequence in context: A378353 A019864 A230151 * A286253 A198674 A168321

Adjacent sequences: A308040 A308041 A308042 * A308044 A308045 A308046

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, May 10 2019

STATUS

approved