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Apéry's Constant Digits

Apéry's constant is defined by(OEIS A002117) where is the Riemann zeta function. was computed to decimal digits by E. Weisstein on Sep. 16, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074).-constant prime occur for , 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... (OEIS A229187).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 89, 211, 2861, 43983, 292702, 8261623, ... (OEIS A036902), which end at digits 23, 457, 7839, 83054, 1256587,..

Apéry's Constant Continued Fraction

The continued fraction for Apéry's constant is [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (OEIS A013631).The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 11, 24, 1, 63, 26, 16, 139, 9, 118, 20, ... (OEIS A229057). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (OEIS A033166), which occur at positions 0, 1, 3, 28, 62, 571, 1555, 2012, 2529, ... (OEIS A229055).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.

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