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Natural Logarithm of 10 Continued Fraction


The continued fraction for ln10 is [0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, ...] (OEIS A016730).

The Engel expansion is 2, 3, 7, 9, 104, 510, 1413, ... (OEIS A059180).

The incrementally largest terms in the continued fraction of ln10 are 2, 3, 6, 26, 716, 774, 982, 1324, 4093, 10322, ... (OEIS A228346), which occur at positions 0, 1, 7, 17, 30, 136, 962, 1163, 1261, 1293, ... (OEIS A228345).

NaturalLogarithmof10ContinuedFractionFirstOccurrences

The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 4, 0, 1, 11, 18, 7, 44, 159, 74, 212, 260, 182, 43, 152, 59, 84, 40, 86, 27, 89, ... (OEIS A228270). The smallest number not occurring in the first 9702786891 terms of the continued fraction are 40230, 45952, 46178, 46530, ... (E. Weisstein, Aug. 18, 2013).

NaturalLogarithmof10KhinchinLevy

Let the continued fraction of ln10 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

Natural Logarithm of 10, Natural Logarithm of 10 Digits

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References

Sloane, N. J. A. Sequences A016730, A059180, A228270, A228345, and A228346 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Natural Logarithm of 10 Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithmof10ContinuedFraction.html

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