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A153810 Decimal expansion of 1 - gamma, where gamma is the Euler's constant (or Euler-Mascheroni constant). 1
4, 2, 2, 7, 8, 4, 3, 3, 5, 0, 9, 8, 4, 6, 7, 1, 3, 9, 3, 9, 3, 4, 8, 7, 9, 0, 9, 9, 1, 7, 5, 9, 7, 5, 6, 8, 9, 5, 7, 8, 4, 0, 6, 6, 4, 0, 6, 0, 0, 7, 6, 4, 0, 1, 1, 9, 4, 2, 3, 2, 7, 6, 5, 1, 1, 5, 1, 3, 2, 2, 7, 3, 2, 2, 2, 3, 3, 5, 3, 2, 9, 0, 6, 3, 0, 5, 2, 9, 3, 6, 7, 0, 8, 2, 5, 3, 2, 5, 0, 4, 8, 5, 3, 6, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Average fractional part of a random (large) integer when divided by all numbers up to it. The result remains true if primes or numbers from particular (fixed) congruence classes are used instead. The result is due to Vallée Poussin. - Charles R Greathouse IV, Apr 11 2012

Expected value of the fractional part of 1/x where x is chosen uniformly at random from (0, 1]. - Charles R Greathouse IV, Apr 11 2012

Value of digamma function psi(x) for x=2. [Stanislav Sykora, Apr 30 2012]

The asymptotic evaluation of the counting function of A064052 ("jagged" numbers) is j(n) ~ log(2)*n - (1-gamma)*n/log(n) + ... - Jean-François Alcover, May 16 2014, after Steven Finch.

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.

LINKS

Table of n, a(n) for n=0..104.

Friedrich Pillichshammer, Euler's constant and averages of fractional parts

Charles Jean de la Vallée Poussin, Sur les valeurs moyennes de certaines fonctions arithmétiques, Annales de la société scientifique de Bruxelles 22 (1898), pp. 84-90.

Wikipedia, Digamma function

FORMULA

Integral from 1 to infinity of {x}dx/x^2, where {x} is the fractional part of x. - Charles R Greathouse IV, Apr 11 2012

Also integral from 0 to infinity of x*log(x)*exp(-x) dx. [Jean-François Alcover, Jun 17 2013]

EXAMPLE

0.422784335...

MATHEMATICA

RealDigits[N[PolyGamma[2], 105]][[1]] (* Arkadiusz Wesolowski, Jan 10 2013 *)

PROG

(PARI) 1-Euler \\ Charles R Greathouse IV, Apr 11 2012

CROSSREFS

Cf. A001620.

Sequence in context: A019834 A087507 A020777 * A098134 A079191 A079184

Adjacent sequences:  A153807 A153808 A153809 * A153811 A153812 A153813

KEYWORD

cons,nonn

AUTHOR

Omar E. Pol, Jan 28 2009

EXTENSIONS

More digits from R. J. Mathar, Feb 06 2009

STATUS

approved

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Last modified October 20 00:14 EDT 2014. Contains 248320 sequences.