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Euler-Mascheroni Constant Continued Fraction


The simple continued fraction of the Euler-Mascheroni constant gamma is [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (OEIS A002852). The first few convergents are 1, 1/2, 3/5, 4/7, 11/19, 15/26, 71/123, 228/395, 3035/5258, 15403/26685, ... (OEIS A046114 and A046115), which are good to 0, 0, 1, 1, 2, 2, 3, 4, 6, 8, 9, 9, 10, ... (OEIS A114541) decimal digits, respectively.

The following table summarizes some record computations of the continued fraction of gamma.

termsdatereference
970258158Sep. 21, 2011E. W. Weisstein
4851382841Jul. 22, 2013E. W. Weisstein
EulerMascheroniConstantContinuedFractionFirstOccurrences

The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 3, 8, 7, 10, 68, 23, 13, 138, 51, 21, ... (OEIS A224847). The smallest positive integers not appearing in the first 4851382841 terms of the continued fraction are 27943, 33436, 33978, 34017, ... (E. W. Weisstein, Jul. 22, 2013).

The sequence of largest terms in the continued fraction is 1, 2, 4, 13, 40, 49, 65, 399, 2076, ... (OEIS A033091), which occur at positions 1, 3, 7, 9, 19, 30, 33, 39, 528, ... (OEIS A224849).

EulerGammaKhinchinLevy

Let the continued fraction of gamma be denoted [a_0;a_1,a_2,...] and let the denominators of the convergents be denoted q_1, q_2, ..., q_n. Then plots above show successive values of a_1^(1/1), (a_1a_2)^(1/2), (a_1a_2...a_n)^(1/n), which appear to converge to Khinchin's constant (left figure) and q_n^(1/n), which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.


See also

Continued Fraction, Euler-Mascheroni Constant, Euler-Mascheroni Constant Digits, Pi Continued Fraction

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References

Sloane, N. J. A. Sequences A002852/M0097, A033091, A046114, A046115, A114541, A224847, and A224849 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Constant Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html

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