A153810 Decimal expansion of 1 - gamma, where gamma is Euler's constant (or the Euler-Mascheroni constant).

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

Offset: 0

Author

Omar E. Pol, Jan 28 2009

Keywords

• cons
• nonn
• nice

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.

Letting eta denote the Dirichlet eta function, and letting zeta denote the Riemann zeta function, we have that 1-gamma is equal to lim x -> infinity 2^x+(4/3)^x-zeta(2-eta(x)). - John M. Campbell, Jan 28 2016

Examples

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References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2000
• Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.2)
• 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.

Crossrefs

Cf. A001620.

Programs

Mathematica

RealDigits[N[PolyGamma[2], 105]][[1]] (* Arkadiusz Wesolowski, Jan 10 2013 *)
RealDigits[1 - EulerGamma, 10, 50][[1]] (* G. C. Greubel, Aug 29 2016 *)

PARI

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

Formula

Equals Integral_{x>=1} {x}dx/x^2 dx, where {x} is the fractional part of x. - Charles R Greathouse IV, Apr 11 2012

Equals Integral_{x>=0} x*log(x)*exp(-x) dx. - Jean-François Alcover, Jun 17 2013

Equals Sum_{n>=2} (zeta(n)-1)/n. - Vaclav Kotesovec, Dec 11 2015

Equals Sum_{k>=1} zeta(2*k+1)/((k+1)*(2*k+1)). - Amiram Eldar, May 24 2021

Equals Sum_{j>=2} Sum_{k>=2} (1/(k * j^k)). - Mike Tryczak, Apr 07 2023

Equals Integral_{x=0..1} {1/x} dx, where {x} is the fractional part of x. From this expression we have 1 - gamma = Sum_{k>=1} Integral_{x=1/(k+1)..1/k} (1/x - k) dx = Sum_{k>=1} (log(1+1/k) - 1/(k+1)). - Jianing Song, Mar 24 2024

Equals Integral_{x>=0} (1/x - 1/(exp(x) - 1))*exp(-x) dx. - Kritsada Moomuang, May 27 2025