Glaisher-Kinkelin Constant Continued Fraction

The continued fraction of  is [1; 3, 1, 1, 5, 1, 1, 1, 3, 12, 4, 1, 271, 1, ...] (OEIS A087501). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.

First occurrences of the terms 1, 2, 3, ... in the continued fraction  occur at , 15, 1, 10, 4, 19, 16, 77, 21, 62, 229, 9, 52, ... (OEIS A225762). The smallest unknown value is 204, which has  (E. Weisstein, Jul. 25, 2013).

The consecutively largest terms are 1, 3, 5, 12, 271, 12574, 13740, 78907, 133430, 574536, ... (OEIS A099791), occurring at positions 0, 1, 4, 9, 12, 266, 3170, 3212, 12961, 82527, ... (OEIS A225752).

Let the continued fraction of  be denoted  and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

REFERENCES:

Sloane, N. J. A. Sequences A087501, A099791, A225752, and A225762 in "The On-Line Encyclopedia of Integer Sequences."