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A307869 Decimal expansion of the asymptotic mean of d(k)/2^omega(k), where d(k) is the number of divisors of k (A000005) and omega(k) is the number of its distinct prime divisors (A001221). 2
1, 4, 2, 7, 6, 5, 6, 5, 3, 5, 4, 2, 4, 8, 3, 9, 8, 8, 3, 1, 1, 7, 5, 2, 3, 9, 3, 9, 6, 8, 7, 3, 2, 7, 9, 0, 4, 0, 9, 3, 7, 3, 3, 6, 2, 8, 0, 7, 4, 4, 3, 9, 2, 7, 4, 2, 2, 4, 7, 4, 1, 4, 3, 6, 7, 3, 4, 4, 2, 9, 8, 8, 3, 4, 1, 1, 5, 3, 8, 9, 4, 0, 7, 4, 8, 3, 0, 3, 5, 2, 6, 0, 8, 3, 7, 4, 0, 5, 1, 7, 7, 9, 3, 2, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

Abdallah Derbal and Meselem Karras, Valeurs moyennes d'une fonction liée aux diviseurs d'un nombre entier, Comptes Rendus Mathematique, Vol. 354, No. 6 (2016), pp. 555-558.‏

Mark Kac, Statistical Independence in Probability, Analysis and Number Theory>/a>, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 79.

FORMULA

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

Equals (Pi^2/6) * Product_{p prime} 1 - 1/(2 * p^2) + 1/(2 * p^3).

EXAMPLE

1.42765653542483988311752393968732790409373362807443...

MATHEMATICA

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{4, -6, 4}, {0, 4, 12}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

CROSSREFS

Cf. A000005, A001221, A034444, A307870.

Sequence in context: A124908 A260593 A143370 * A016695 A125271 A245262

Adjacent sequences:  A307866 A307867 A307868 * A307870 A307871 A307872

KEYWORD

nonn,cons,changed

AUTHOR

Amiram Eldar, May 02 2019

EXTENSIONS

More terms from Vaclav Kotesovec, May 29 2020

STATUS

approved

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Last modified June 1 19:32 EDT 2020. Contains 334762 sequences. (Running on oeis4.)