TOPICS
Search

Soldner's Constant Continued Fraction

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The continued fraction for mu is given by [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...] (OEIS A099803).

SoldnerssConstantContinuedFractionFirstOccurrences

The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 1, 6, 2, 47, 28, 21, 107, 114, ... (OEIS A000000).

The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... (OEIS A099804), which occur at positions 0, 1, 2, 11, 69, 125, 201, 584, 1591, 2435, ... (OEIS A229230).

SoldnerKhinchinLevy

Let the continued fraction of mu 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

Soldner's Constant, Soldner's Constant Digits

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A099803, A099804, and A229230 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Soldner's Constant Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SoldnersConstantContinuedFraction.html

Subject classifications