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A153810
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Decimal expansion of 1 - gamma, where gamma is Euler's constant (or the Euler-Mascheroni constant).
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7
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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
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OFFSET
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0,1
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COMMENTS
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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
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
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
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LINKS
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FORMULA
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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
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EXAMPLE
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0.422784335...
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MATHEMATICA
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RealDigits[1 - EulerGamma, 10, 50][[1]] (* G. C. Greubel, Aug 29 2016 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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