Simple Continued Fraction Expansion of Khinchin's Constant

In 1935, A.Ya. Khinchin showed that "the geometric mean of the first n elements [of almost any irrational number's simple continued fraction] approaches the absolute constant 2.6..." This number is now known as Khinchin's constant. In May 1998, Xavier Gourdon computed 110,000 digits of this number. Mathematica 4 converts this approximation to 106,621 terms of the constant's simple continued fraction form. Dropping the last two of these terms, we have (or, take me past the number):

cfKhinchin = [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, 1, 1, 90, 2, 1, 12, 1, 1, 1, 1, 5, 2, 6, 1, 6, 3, 1, 1, 2, 5, 2, 1, 2, 1, 1, 4, 1, 2, 2, 3, 2, 1, 1, 4, 1, 1, 2, 5, 2, 1, 1, 3, 29, 8, 3, 1, 4, 3, 1, 10, 50, 1, 2, 2, 7, 6, 2, 2, 16, 4, 4, 2, 2, 3, 1, 1, 7, 1, 5, 1, 2, 1, 5, 3, 1, 1, 1, 2, 2, 2, 1, 13, 11, 770, 1, 4, 2, 1, 14, 1, 14, 2, 1, 6, 1, 1, 1, 9, 2, 53, 1, 2, 2, 1, 9, 5, 6, 2, 1, 2, 1, 5, 4, 1, 234, 7, 1, 1, 4, 3, 19, 3, 1, 10, 18, 8, 24, 1, 12, 1, 1, 10, 3, 2, 1, 32, 112, 5, 1, 1, 3, 2, 5, 1, 2, 1, 3, 2, 1, 2, 1, 1, 2, 2, 4, 1, 6, 4, 1, 2, 1, 8, 2, 1, 4, 2, 1, 1, 11, 1, 1, 1, 5, 3, 4, 2, 6, 2, 1, 2, 1, 1, 19, 1, 38, 2, 1, 1, 4, 6, 2, 50, 2, 1, 1, 2, 1, 4, 1, 5, 1, 2, 8, 13, 1, 2, 1, 1, 9, 1, 6, 3, 6, 1, 4, 2, 1, 272, 1, 1, 1, 1, 4, 1, 21, 3, 1, 2, 87, 1, 8, 1, 2, 3, 2, 1, 1, 2, 3, 16, 1, 5, 3, 5, 1, 1, 1, 10, 11, 45, 2, 331, 2, 1, 2, 1, 4, 1, 2, 2, 1, 3, 1, 1, 3, 1, 2, 2, 1, 13, 1, 3, 3, 2, 4, 4, 1, 4, 40, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 1, 6, 12, 6, 1, 2, 1, 5, 4, 1, 2, 2, 1, 1, 1, 7, 5, 2, 9, 12, 6, 4, 1, 2, 2, 1, 7, 3, 3, 5, 10, 1, 2, 13, 1, 7, 3, 4, 2, 5, 1, 1, 2, 2, 2, 3, 1, 1, 14, 121, 1, 1, 2, 1, 1, 5, 1, 47, 20, 1, 1, 2, 1, 4, 1, 9, 5, 3, 1, 3, 2, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 1, 1, 3, 2, 1, 16, 3, 4, 3, 10, 2, 5, 2, 2, 5, 1, 12, 33, 2, 1, 7, 1, 3, 1, 4, 10, 1, 1, 2, 1, 9, 1, 4, 4, 14, 2, 1, 2, 12, 1, 19, 1, 1, 1, 1, 1, 10, 1, 1, 4, 1, 1, 3, 1, 2, 1, 24, 1, 1, 3, 1, 5, 2, 1, 2, 13, 2, 3, 2, 1, 2, 1, 5, 2, 2, 2, 3, 1, 2, 6, 2, 1, 2, 2, 16, 6, 6, 10, 1, 1, 1, 1, 2, 10, 24, 1, 1, 2, 1, 2, 4, 1, 1, 2, 4, 1, 10, 3, 1, 9, 1, 6, 1, 1, 34, 1, 3, 1, 4, 1, 3, 2, 11, 2, 5, 1, 1, 1, 2, 1, 16, 46, 3, 5, 1, 2, 2, 6, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 2, 1, 2, 2, 1, 9, 1, 4, 2, 1, 2, 1, 1, 3, 12, 13, 34, 1, 1, 84, 1, 4, 2, 4, 1, 1, 2, 1, 3, 2, 24, 1, 2, 2, 1, 1, 1, 1, 7, 1, 4, 1, 2, 1, 22, 2, 1, 8, 1, 4, 1, 5, 1, 3, 3, 8, 2, 54, 2, 4, 2, 1, 1, 1, 4, 1, 8, 1, 9, 2, 1, 7, 18, 7, 2, 3, 381, 1, 3, 3, 5, 4, 1, 9, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 3, 2, 1, 5, 2, 14, 1, 2, 7, 1, 3, 3, 3, 1, 5, 1, 2, 2, 1, 1, 1, 1, 3, 10, 1, 3, 1, 6, 1, 1, 2, 3, 1, 5, 1, 4, 1, 2, 1, 15, 1, 15, 1, 23, 1, 2, 4, 5, 1, 1, 1, 9, 2, 2, 5, 1, 5, 1, 1, 1, 1, 4, 8, 1, 2, 1, 1, 1, 1, 9, 1, 3, 1, 1, 8, 2, 1, 1, 1, 1, 10, 1, 1, 1, 2, 7, 1, 13, 7, 2, 1, 5, 1, 5, 1, 3, 2, 3, 7, 12, 2, 18, 4, 6, 1, 4, 38, 2, 11, 1, 5, 5, 3, 1, 5, 1, 5, 1, 2, 1, 3, 1, 1, 1, 144, 2, 1, 7, 2, 1, 1, 2, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 6, 3, 18, 3, 2, 2, 1, 1, 1, 2, 2, 6, 2, 2, 1, 1, 1, 8, 3, 31, 1, 15, 1, 22, 1, 1, 1, 2, 1, 3, 2, 1, 10, 3, 4, 45, 1, 2, 1, 1, 5, 3, 15, 2, 2, 1, 4, 1, 1, 1, 3, 1, 2, 12, 1, 1, 1, 6, 2, 12, 1, 1, 1, 5, 3, 5, 1, 4, 1, 4, 1, 1, 4, 3, 2, 1, 10, 1, 12, 3, 1, 1, 15, 1, 2, 5, 1, 2, 1, 89, 6, 15, 2, 5, 8, 5, 4, 3, 1, 4, 1, 1, 3, 7, 1, 2, 3, 1, 19, 7, 1, 1, 1, 5, 1, 1, 1, 1, 9, 4, 3, 1, 3, 7, 3, 9, 1, 1, 2, 21, 1, 1, 9, 8, 2, 1, 1, 1, 1, 6, 11, 12, 1, 1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 7, 4, 10, 2, 7, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 5, 1, 1, 3, 1, 1, 2, 1, 7, 1, 1, 1, 1, 2, 1, 4, 1, 3, 46, 51, 1, 15, 15, 5, 12, 21, 7, 1, 5, 7, 1, 6, 8, 1, 4, 3, 1, 9, 1, 2, 5, 1, 2, 4, 1, 1, 2, 1, 8, 2, 3, 8, 1, 2, 2, 1, 1, 1, 9, 6, 1, 8, 3, 24, 7, 1, 163, 1, 2, 6, 27, 2, 1, 58, 1, 1, 1, 9, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 11, 3, 1, 3, 3, 1, 2, 1, 1, 2, 4, 2, 21, 1, 4, 1, 1, 2, 1, 3, 1, 2, 2, 1, 5, 4, 1, 5, 1, 509, 2, 4, 3, 1, 3, 1, 18, 1, 10, 17, 2, 8, 1, 12, 1, 4, 8, 16, 7, 8, 1, 1, 2, 2, 12, 3, 1, 1, 11, 6, 38, 1, 1, 1, 1, 2, 1, 1, 5, 1, 3, 1, 1, 1, 2, 13, 22, 1, 1, 2, 1, 1, 2, 9, 1, 12, 1, 7, 1, 4, 1, 1, 1, 1, 1, 6, 5, 1, 27, 2, 1, 2, 1, 2, 1, 2, 2, 2, 4, 2, 1, 1, 2, 1, 3, 1, 6, 1, 1, 9, 1, 22, 6, 3, 1, 3, 4, 5, 9, 1, 7, 1, 4, 5, 8, 2, 2, 1, 2, 1, 2, 1, 1, 11, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 3, 3, 87, 1, 3, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 2, 10, 2, 104, 1, 2, 1, 4, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 6, 1, 2, 7, 1, 4, 1, 10, 1, 1, 2, 4, 1, 2, 1, 9, 12, 2, 3, 2, 4, 3, 1, 1, 1, 1, 2, 1, 11, 1, 2, 2, 3, 45, 3, 1, 1, 2, 3, 2, 1, 23, 1, 2, 9, 1, 3, 55, 1, 2, 2, 2, 1, 2, 1, 1, 1, 3, 5, 74, 2, 11, 1, 1, 12, 1, 6, 1, 2, 4, 1, 6, 1, 3, 4, 2, 139, 9, 1, 3, 1, 3, 10, 2, 1, 6, 2, 4, 6, 6, 1, 1, 55, 5, 40, 60, 1, 1, 4, 1, 3, 1, 1, 1, 8, 1, 2, 1, 30, 18, 1, 13, 3, 17, 1, 13, 10, 16, 3, 3, 1, 5, 2, 2, 14, 1, 1, 24, 57, 1, 2, 2, 14, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 7, 1, 4, 1, 15, 152, 2, 7, 1, 17, 1, 4, 16, 2, 11, 2, 1, 3, 2, 3, 1, 2, 1, 1, 2, 4, 12, 1, 35, 1, 1, 1, 2, 3, 12, 1, 8, 1, 1, 2, 1, 8, 3, 10, 4, 3, 1, 1, 2, 1, 25, 4, 3, 4, 1, 15, 1, 1, 7, 1, 3, 1, 1, 132, 2, 1, 1, 37, 1, 60, 2, 5, 1, 1, 2, 2, 11, 12, 2, 1, 1, 1, 1, 3, 1, 3, 1, 4, 1, 4, 4, 2, 1, 7, 1, 4, 17, 1, 1, 6, 1, 2, 1, 4, 1, 47, 2, 4, 1, 3, 1, 3, 1, 1, 1, 1, 52, 3, 6, 5, 1, 1, 1, 1, 1, 3, 96, 5, 3, 116, 1, 5, 2, 1, 4, 2, 11, 1, 1, 6, 14, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 2, 2, 38, 1, 16, 1, 2, 1, 1, 1, 2, 1, 6, 1, 6, 1, 21, 3, 5, 2, 2, 1, 2, 3, 3, 1, 2, 5, 7, 2, 2, 1, 1, 2, 6, 1, 56, 3, 23, 2, 16, 4, 1, 2, 6, 2, 2, 3, 4, 4, 7, 2, 1, 1, 1, 3, 3, 4, 4, 1, 2, 3, 6, 2, 3, 1, 1, 3, 5, 9, 4, 1, 8, 2, 3, 1, 2, 1, 1, 1, 1, 14, 2, 6, 15, 2, 26, 1, 2, 5, 3, 1, 28, 1, 84, 4, 6, 1, 4, 1, 3, 10, 2, 1, 2, 1, 4, 1, 1, 4, 1, 1, 1, 4, 8, 11, 2, 439, 5, 34, 10, 1, 2, 1, 2, 35, 1, 3, 3, 1, 2, 2, 4, 53, 3, 2, 18, 1, 2, 6, 2, 1, 11, 1, 5, 1, 2, 3, 2, 72, 1, 1, 4, 2, 2, 2, 1, 34, 1, 1, 5, 3, 1, 1, 2, 2, 1, 941, 1, 3, 7, 1, 4, 2, 5, 1, 3, 2, 3, 1, 18, 1, 1, 2, 4, 1, 2, 3, 1, 3, 7, 3, 1, 1, 35, 1, 104, 9, 7, 1, 1, 2, 2, 2, 3, 1, 4, 4, 1, 31, 4, 12, 1, 1, 5, 7, 9, 5, 1, 1, 10, 1, 6, 2, 2, 1, 1, 1, 1, 2, 9, 4, 2, 1, 11, 2, 3, 1, 2, 4, 1, 1, 4, 2, 128, 3, 10, 1, 36, 2, 2, 1, 10, 1, 9, 3, 1, 37, 1, 1, 3, 12, 6, 1, 1, 4, 1, 14, 3, 1, 12, 5, 1, 1, 2, 4, 1, 11, 1, 14, 1, 2, 1, 1, 80, 6, 6, 8, 322, 1, 1, 1, 5, 2, 1, 1, 2, 5, 1, 1, 8, 38, 1, 6, 7, 2, 6, 1, 3, 5, 23, 2, 1, 1, 5, 1, 1, 3, 2, 1, 1, 2, 1, 1, 7, 4, 4, 4, 1, 4, 37, 1, 1, 1, 7, 2, 301, 3, 1, 3, 1, 3, 1, 1, 1, 2, 2, 3, 6, 3, 2, 1, 1, 2, 1, 2, 1, 4, 1, 29, 22, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 9, 2, 1, 3, 2, 8, 1, 6, 23, 1, 2, 1, 3, 32, 1, 1, 1, 2, 7, 1, 1, 6, 6, 2, 3, 12, 5, 1, 1, 1, 12, 1, 9, 1, 13, 1, 3, 1, 5, 6, 7, 15, 11, 4, 47, 2, 1, 13, 1, 3, 4, 8, 3, 4, 4, 2, 2, 5, 1, 18, 1, 4, 3, 3, 1, 2, 184, 1, 4, 2, 2, 3, 2, 1, 6, 1, 2, 33, 45, 1, 2, 2, 5, 6, 1, 1, 108, 458, 1, 1, 1, 11, 8, 4, 1, 4, 1, 1, 1, 1, 1, 2, 1, 3, 1, 8, 5, 1, 89, 5, 9, 6, 9, 1, 5, 1, 3, 2, 1, 3, 1, 19, 1, 1, 2, 5, 2, 28, 1, 42, 2, 1, 5, 3, 3, 1, 5, 20, 2, 3, 3, 3, 1, 1, 2, 4, 3, 1, 1, 14, 1, 2, 1, 8, 3, 1, 2, 5, 2, 1, 1, 3, 4, 3, 4, 8, 7, 3, 6, 20, 19, 1, 83, 1, 1, 2, 8, 1, 4, 2, 1, 1, 132, 11, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 80, 2, 5, 1, 4, 2, 34, 1, 11, 1, 1, 11, 1, 12, 3, 3, 1, 1, 1, 5, 1, 2, 2, 1, 3, 3, 4, 5, 2, 1, 1, 1, 6, 3, 2, 27, 1, 7, 2, 3, 4, 1, 1, 2, 4, 1, 2, 1, 9, 7, 15, 7, 2, 5, 3, 3, 8, 1, 1, 2, 1, 1, 59, 2, 2, 4, 1, 15, 1, 3, 2, 9, 1, 3, 13, 9, 29, 1, 14, 4, 51, 15, 12, 1, 8, 203, 2, 1, 6, 1, 2, 1, 2, 184, 1, 1, 2, 3, 3, 14, 3, 1, 2, 2, 10, 2, 1, 18, 4, 1, 4, 2, 1, 3, 4, 5, 2, 2, 15, 11, 3, 1, 1, 5, 28, 3, 10, 1, 4, 6, 1, 1, 38, 1, 2, 1, 1, 1, 6, 1, 1, 1, 4, 8, 1, 4, 1, 1, 2, 2, 4, 4, 1, 49, 1, 1, 2, 3, 1, 3, 1, 497, 1, 9, 1, 2, 1, 3, 1, 1, 9, 1, 5, 2, 6, 123, 1, 1, 1, 2, 29, 3, 6, 1, 1, 1, 3, 1, 4, 1, 464, 3, 1, 6, 1, 6, 1, 9, 4, 1, 3, 1, 3, 2, 1, 1, 8, 1, 1, 1, 2, 1, 79, 9, 5, 3, 1, 1, 1, 1, 4, 3, 5, 1, 33, 14, 2, 2, 2, 2, 3, 20, 2, 1, 1, 1, 3, 2, 37, 53, 1, 1, 2, 1, 1, 12, 1, 4, 1, 1, 1, 1, 92, 2, 4, 3, 82, 1, 15, 3, 5, 1, 1, 17, 1, 18, 2, 1, 2, 1, 4, 3, 14, 1, 1, 7, 1, 1, 2, 2, 53, 1, 2, 1, 2, 1, 1, 1, 2, 24, 3, 4, 2, 1, 3, 5, 5, 1, 1, 1, 2, 2, 1, 8, 4, 2, 100, 1, 1, 4, 2, 1, 6, 1, 7, 1, 1, 1, 1, 2, 1, 2, 1, 11, 10, 2, 1, 1, 3, 3, 2, 4, 2, 11, 1, 5, 1, 2, 1, 4, 1, 2, 1, 2, 11759, 1, 57, 1, 186, 7, 6, 2, 1, 4, 3, 1, 5, 1, 8, 2, 2, 1417, 2, 1, 4, 17, 1, 1, 1, 1, 2, 8, 1, 8, 1, 7, 24, 20, 1, 1, 1, 76, 2, 2, 1, 4, 1, 1, 18, 3, 2, 2, 1, 1, 1, 8, 2, 64, 2, 1, 2, 5, 1, 26, 1, 1, 1, 5, 7, 2, 1, 31, 9, 2, 2, 1, 1, 509, 1, 2, 1, 1, 4, 1, 3, 6, 1, 1, 1, 6, 1, 4, 1, 3, 1, 2, 1, 1, 2, 14, 1, 16, 1, 5, 1, 5, 1, 5, 1, 2, 1, 2, 1, 2, 1, 5, 1, 4, 1, 51, 2, 1, 18, 1, 3, 2, 11, 1, 4, 7, 8, 1, 1, 1, 87, 4, 1, 6, 1, 1, 1, 3, 1, 1, 3, 4, 2, 1, 5, 1, 10, 7, 1, 2, 6, 1, 5, 1, 1, 9, 3, 1, 1, 2, 1, 4, 1, 2, 135, 8, 1, 1, 2, 1, 1, 1, 8, 1, 2, 5, 1, 12, 2, 4, 3, 1, 1, 130, 3, 1, 3, 1, 1, 1, 2, 1, 11, 5, 1, 1, 3, 4, 1, 3, 18, 2, 3, 2, 3, 12, 1, 8, 1, 5, 1, 144, 1, 1, 1, 2, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 2, 1, 3, 1, 2, 1, 4, 3, 4, 41, 4, 4, 2, 7, 12, 1, 1, 1, 2, 1, 17, 17, 3, 17, 1, 1, 2, 1, 9, 1, 2, 3, 3, 2, 1, 1, 10, 2, 2, 41, 1, 5, 1, 1, 3, 3, 8, 1, 3, 1, 2, 4, 1, 2, 4, 5, 1, 13, 1, 6, 6, 5, 1, 2, 2, 1, 4, 1, 3, 7, 1, 1, 1, 1, 3, 2, 6, 12, 1, 6, 1, 4, 5, 2, 1, 8, 2, 1, 1, 2, 5, 5, 3, 2, 1, 3, 6, 2, 1, 1, 3, 1, 1, 12, 29, 20, 2, 15, 1, 1, 3, 1, 1, 1, 1, 3, 25, 5, 1, 17, 12, 2, 1, 1, 1, 1, 2, 2, 1, 6, 1, 4, 1, 10, 11, 5, 2, 1, 3, 2, 2, 1, 5, 2, 1, 1, 1, 4, 4, 1, 1, 8, 2, 4, 3, 3, 3, 1, 15, 1, 1, 4, 2, 1, 13, 1, 20, 1, 2, 144, 2, 2, 1, 7, 1, 1, 4, 1, 23, 1, 1, 11, 3, 3, 1, 1, 2, 3, 2, 5, 1, 10, 3, 2, 1, 2, 1, 1, 5, 3, 1, 5, 1, 1, 2, 21, 1, 1, 1, 5, 1, 35, 6, 1, 16, 1, 1, 1, 1, 7, 4, 1, 7, 2, 1, 1, 2, 1, 2, 8, 122, 1, 2, 11, 1, 2, 11, 1, 3, 5, 1, 2, 1, 2, 1, 2, 9, 1, 6, 1, 10, 1, 3, 2, 2, 1, 1, 1, 1, 3, 2, 1, 1, 6, 2, 26, 2, 24, 1, 1, 16, 1, 1, 19, 4, 4, 3, 2, 4, 5, 6, 1, 1, 3, 2, 2, 2, 5, 1, 1, 41, 2, 149, 2, 178, 1, 1, 4, 2, 1, 1, 54, 1, 4, 2, 6, 10, 1, 1, 2, 3, 1, 161, 8, 1, 2, 20, 2, 16, 1, 1, 1, 1, 1, 1, 19, 3, 1, 5, 1, 5, 3, 2, 16, 1, 4, 4, 11, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 4, 1, 11, 1, 4595, 1, 1, 1, 4, 1, 1, 3, 10, 5, 29, 3, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 6, 3, 2, 26, 1, 2, 1, 1, 4, 3, 1, 2, 1, 5, 2, 4, 1, 1, 1, 1, 2, 2, 3, 28, 1, 3, 2, 1, 7, 3, 1, 4, 2, 5, 15, 2, 1, 7, 1, 5, 4, 2, 16, 1, 3, 6, 1, 2, 4, 1, 90, 1, 1, 1, 1, 4, 2, 1, 4, 3, 1, 12, 1, 1, 22, 3, 1, 2, 7, 10, 4, 3, 4, 21, 5, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 6, 5, 1, 1, 10, 1, 6, 2, 3, 3, 2, 5, 1, 39, 1, 3, 1, 3, 1, 6, 1, 1, 1, 16, 2, 1, 6, 1, 3, 1, 1, 2, 2, 3, 1, 3, 4, 2, 1, 3, 1, 1, 4, 1, 1, 2, 27, 7, 4, 1, 6, 1, 2, 1, 2, 1, 3, 5, 1, 1, 3, 3, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 10, 1, 7, 1, 1, 1, 4, 5, 1, 1, 1, 1, 11, 1, 1, 5, 3, 1, 2, 4, 1, 2, 1, 3, 4, 1, 1, 2, 2, 4, 3, 1, 1, 2, 12, 4, 2, 4, 1, 3, 1, 7, 2, 1, 2, 2, 17, 3, 1, 3, 1, 1, 35, 1, 1, 1, 1, 1, 3, 1, 12, 2, 29, 1, 7, 7, 1, 6, 1, 7, 2, 2, 1, 5, 2, 3, 4, 13, 2, 11, 2, 1, 21, 113, 22, 1, 1, 1, 1, 3, 8, 4, 2, 1, 1, 1, 1, 82, 2, 1, 16, 1, 1, 10, 1, 1, 2, 1, 1, 2, 4, 1, 7, 23, 22, 2, 1, 1, 4, 1, 1, 13, 119, 1, 4, 1, 2, 1, 1, 10, 1, 1, 172, 1, 2, 82, 1, 1, 3, 196, 3, 4, 1, 2, 16, 1, 4, 2, 1, 11, 3, 27, 1, 56, 2, 4, 2, 1, 1, 2, 1, 3, 2, 5, 1, 1, 34, 2, 1, 2, 1, 17, 1, 1, 10, 1, 1, 2, 2, 6, 1, 1, 8, 2, 2, 9, 1, 131, 1, 3, 1, 1, 1, 1, 5, 10, 8, 2, 5, 5, 42, 1, 7, 1, 6, 1, 11, 1, 1, 78, 3, 1, 6, 1, 1, 1, 5, 18, 1, 10, 8, 10, 2, 31, 3, 1, 25, 2, 1, 1, 5, 5, 1, 2, 6, 5, 3, 4, 3, 2, 3, 1, 2, 3, 25, 6, 1, 1, 161, 1, 1, 20, 1, 5, 30, 2, 22, 1, 7, 4, 1, 116, 1, 3, 45, 4, 4, 3, 2, 1, 146, 21, 2, 1, 9, 1, 6, 2, 1, 2, 1, 1, 6, 1, 10, 5, 2, 6, 16, 1, 1, 1, 11, 6, 11, 6, 1, 1, 2, 9, 3, 1, 2, 2, 2, 1, 1, 1, 8, 1, 8, 2, 1, 3, 2, 2, 3, 1, 1, 5, 10, 1, 1, 1, 36, 1, 1, 14, 3, 1, 10, 1, 1, 1, 1, 1, 37, 5, 5, 5, 9, 1, 274, 1, 5, 1, 3, 2, 1, 2, 5, 2, 1, 1, 1, 1, 2, 1, 1, 10, 6, 3, 2, 89, 2, 5, 1, 1, 7, 1, 2, 1, 13, 2, 8, 1, 3, 1, 6, 1, 6, 1, 14, 2, 1, 1, 1, 6, 2, 1, 4, 3, 1, 5, 3, 1, 11, 47, 1, 1, 1, 7, 3, 6, 53, 8, 2, 18, 23, 15, 1, 12, 1, 1, 3, 7, 1, 2, 1, 3, 1, 1, 3, 21, 1, 1, 8, 2, 1, 2, 14, 2, 1, 3, 1, 8, 1, 2, 10, 13, 2, 3, 3, 1, 32, 2, 1, 5, 5, 16, 2, 4, 2, 2, 1, 1, 14, 1, 1, 7, 5, 1, 10, 1, 3, 1, 1, 21, 11, 5, 1, 1, 1, 4, 1, 5, 6, 3, 9, 3, 3, 14, 3, 65, 1, 7, 1, 15, 2, 12, 1, 1, 3, 2, 1, 2, 1, 2, 5, 1, 70, 4, 13, 5, 78, 1, 4, 2, 4, 5, 3, 5, 10, 2, 1, 4, 8, 1, 1, 1, 3, 2, 2, 1, 1, 822, 2, 6, 1, 5, 2, 1, 4, 1, 3, 1, 2, 2, 29, 2, 1, 12, 1, 1, 4, 6, 6, 2, 2, 3, 1, 1, 118, 1, 22, 2, 23, 2, 1, 1, 1, 7, 1, 5, 1, 6, 5, 2, 12, 9, 1, 2, 2, 18, 163, 1, 1, 6, 3, 1, 1, 1, 1, 2, 2, 17, 1, 2, 4, 3, 7, 1, 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5, 3, 3, 9, 2, 1, 1, 3, 15, 1, 2, 7, 8, 1, 9, 1, 1, 1, 1, 2, 3, 8, 2, 1, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 2, 1, 3, 6, 1, 1, 2, 1, 2, 1, 45, 1, 1, 2, 5, 1, 1, 1, 87, 1, 477, 1, 1, 4, 1, 4, 5, 1, 13, 2, 2, 3, 1, 1, 7, 11, 5, 8, 2, 1, 1, 1, 2, 1, 1, 2, 2, 7, 3, 1, 14, 1, 3, 1, 1, 2, 1, 1, 2, 51, 4, 2, 14, 3, 1, 32, 1, 1, 14, 2, 4, 5, 2, 61, 2, 16, 4, 1, 6, 3, 3, 1, 1, 1, 1, 61, 1, 29, 1, 8, 1, 2, 1, 1, 5, 1, 2, 1, 14, 2, 2, 1, 1, 1, 1, 4, 3, 1, 1, 3, 3, 2, 1, 2, 2, 14, 2, 8, 1, 3, 18, 15, 1, 13, 1, 2, 1, 2, 3, 8, 11, 1, 3, 1, 2, 1, 6, 3, 3, 1, 5, 2, 5, 1, 1, 6, 4, 1, 2, 22, 2, 1, 6, 1, 1, 3, 6, 14, 2, 10, 1, 3, 1, 1, 1, 3, 1, 1, 6, 1, 1, 140, 3, 48, 2, 8, 7, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 38, 1, 2, 1, 1, 1, 3, 21, 32, 5, 2, 2, 1, 4, 1, 1, 6, 2, 4, 2, 128, 3, 7, 2, 1, 18, 11, 4, 1, 1, 2, 1, 7, 1, 33, 3, 2, 1, 6, 1, 1, 1, 1, 5, 4, 3, 4, 9, 2, 1, 1, 7, 1, 13, 2, 79, 1, 30, 1, 1, 1, 1, 1, 8, 13, 1, 15, 30, 2, 1, 1, 27, 3, 1, 6, 3, 4, 3, 1, 1, 5, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 13, 3, 2, 5, 65, 8, 1, 11, 3, 1, 2, 2, 4, 36, 1, 8, 1, 6, 2, 9, 3, 5, 1, 2, 2, 1, 3, 5, 3, 51, 1, 4, 2, 3, 2, 1, 1, 16, 19, 1, 52, 2, 1, 7, 20, 3, 2, 2, 1, 6, 35, 2, 1, 1, 5, 4, 2, 3, 1, 5, 1, 3, 1, 1, 96, 1, 28, 3, 1, 1, 2, 2, 96, 1, 3, 8, 1, 1, 1, 2, 2, 1, 4, 1, 7, 1, 1, 1, 26, 1, 1, 650, 1, 2, 2, 1, 3, 1, 7, 3, 54, 2, 2, 3, 1, 1, 1, 3, 1, 2, 4, 1, 3, 1, 1, 243, 1, 5, 1, 5, 2, 1, 1, 1, 1, 1, 1, 4, 3, 1, 3, 1, 4, 2, 1, 1, 2, 1, 1, 2, 167, 38, 1, 1, 9, 1, 13, 2, 1, 1, 3, 3, 1, 2, 2, 29, 5, 167, 11, 1, 47, 1, 4, 1, 16, 2, 2, 1, 1, 1, 1, 1, 11, 332, 1, 2, 1, 2, 2, 10, 2, 47, 4, 2, 9, 1, 22, 2, 8, 1, 1, 24, 4, 1, 4, 1, 1, 1, 1, 12, 4, 3, 30, 1, 3, 9, 2, 1, 3, 11, 1, 1, 4, 1, 5, 1, 2, 30, 1, 97, 1, 1, 10, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 68, 1, 2, 1, 2, 1, 2, 4, 3, 13, 1, 1, 8, 1, 4, 1, 2, 1, 16, 14, 1, 4, 2, 2, 2, 32, 2, 12, 3, 2, 9, 1, 2, 2, 1, 2, 2, 2, 3, 3, 27, 1, 103, 1, 2, 22, 2, 1, 12, 1, 1, 7, 2, 1, 4, 1, 1, 3, 12, 2, 7, 6, 3, 16, 3, 5, 4, 1, 2, 5, 4, 9, 8, 1, 18, 3, 2, 1, 6, 1, 39, 3, 2, 3, 15, 5, 1, 2, 1, 7, 17, 2, 1, 3, 1, 246, 2, 3, 3, 38, 1, 4, 1, 10, 1, 2, 15, 1, 2, 1, 2, 2, 74, 2, 4, 3, 2, 2, 2, 9, 1, 5, 1, 3, 3, 5, 1, 1, 14, 9, 1, 103, 83, 3, 34, 1, 1, 1, 2, 14, 15, 5, 9, 1, 50, 1, 1, 1, 1, 6, 4, 38, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 2, 4, 4, 1, 1, 1, 2, 3, 2, 2, 9, 1, 2, 1, 17, 2, 1, 1, 12, 1, 1, 1, 4, 1, 9, 1, 1, 1, 1, 9, 1, 8, 1, 7, 1, 7, 1, 1, 2, 6, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7, 2, 1, 6, 13, 13, 1, 11, 1, 2, 2, 1, 1, 1, 10, 5, 6, 1, 1, 32, 2, 15, 8, 1, 6, 3, 2, 3, 13, 1, 2, 1, 1, 1, 2, 38, 2, 68, 9, 6, 2, 14, 1, 2, 4, 19, 1, 10, 7, 2, 4, 1, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 45, 2, 11, 3, 9, 13, 2, 1, 2, 56, 3, 1, 1, 1, 2, 2, 5, 1, 149, 1, 1, 17, 13, 2, 1, 22, 1, 1, 13, 1, 39, 1, 4, 1, 6, 5, 2, 2, 2, 1, 4, 2, 1, 6, 1, 7, 2, 3, 1, 5, 3, 1, 1, 5, 4, 1, 2, 1, 1, 1, 4, 3, 2, 8, 17, 1, 1, 1, 2, 3, 3, 36, 2, 2, 3, 3, 4, 11, 1, 3, 2, 1, 5, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 3, 77, 5, 2, 1, 14, 3, 1, 1, 2, 1, 1, 3, 1, 1, 10, 1, 1, 68, 1, 2, 2, 2, 1, 35, 1, 1, 2, 6, 1, 1, 1, 1, 11, 1, 5, 1, 2, 2, 11, 1, 3, 1, 1, 1, 5, 3, 2, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 4, 1, 1, 2, 1, 9, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 3, 4, 1, 5, 1, 1, 1, 2, 1, 1, 2, 2, 1, 40, 2, 2, 18, 5, 1, 2, 1, 10, 3, 5, 1, 2, 1, 3, 4, 1, 2, 3, 6, 1, 1, 1, 20, 3, 1, 4, 3, 9, 3, 1, 5, 1, 1, 9, 1, 2, 1, 27, 1, 8, 5, 7, 1, 9, 11, 1, 24, 2, 1, 1, 1, 3, 5, 1, 6, 20, 2, 2, 1, 7, 1, 1, 1, 9, 2, 2, 1, 2, 1, 1, 2, 7, 7, 1, 1, 1, 2, 6, 1469, 1, 2, 13, 4, 2, 4, 1, 1, 6, 3, 6, 1, 21, 1, 1, 1, 9, 1, 2, 15, 4, 2, 7, 1, 1, 1, 1, 4, 27, 3, 3, 74, 6, 75, 1, 4, 2, 1, 1, 3, 1, 2, 3, 2, 1, 52, 2, 75, 1, 7, 5, 11, 4, 20, 2, 2, 1, 1, 2, 20, 16, 2, 4, 1, 1, 6, 1, 1, 4, 1, 8, 2, 8, 19, 3, 1, 1, 40, 1, 2, 2, 23, 2, 1, 2, 4, 3, 1, 1, 24, 1, 3, 1, 1, 4, 2, 1, 5, 1, 2, 1, 24, 2, 1, 4, 12, 5, 4, 1, 4, 4, 1, 9, 2, 2, 1, 9, 1, 1, 3, 1, 1, 7, 3, 1, 4, 1, 1, 1, 10, 13, 5, 1, 2, 4, 1, 5, 2, 1, 1, 2, 1, 2, 1, 3, 2, 1, 8, 1, 1, 3, 1, 1, 1, 33, 16, 1, 1, 1, 7, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 5, 1, 1, 11, 1, 3, 2, 1, 2, 2, 1, 1, 88, 2, 4, 4, 1, 1, 3, 4, 2, 3, 1, 13, 6, 79, 13, 2, 3, 2, 1, 2, 9, 36, 2, 1, 3, 5, 3, 1, 1, 1, 1, 2, 1, 29, 1, 1, 3, 81, 5, 3, 1, 11, 1, 2, 231973, 15, 13, 2, 116, 2, 2, 1, 7, 32, 1, 1, 1, 2, 11, 1, 1, 1, 21, 1, 1, 13, 1, 4, 34, 2, 11, 3, 1, 1, 14, 1, 2, 58, 1, 23, 1, 21, 4, 2, 13, 2, 2, 899, 1, 2, 34, 1, 2, 2, 5, 1, 4, 32, 1, 5, 2, 2, 2, 9, 1, 3, 2, 1, 4, 2, 1, 7, 3, 11, 4, 2, 1, 1, 122, 2, 1, 8, 3, 2, 1, 21, 1, 7, 1, 18, 1, 4, 92, 1, 3, 2, 1, 1, 4, 1, 1, 4, 2, 9, 22, 1, 1, 3, 2, 1, 40, 5, 65, 14, 7, 2, 2, 2, 13, 1, 1, 1, 1, 48, 1, 7, 1, 19, 1, 3, 1, 1, 5, 2, 3, 1, 6, 1, 1, 10, 4, 1, 1, 3, 1, 1, 5, 1, 2, 1, 1, 1, 9, 1, 7, 1, 1, 1, 2, 1, 8, 1, 6, 1, 12, 12, 23, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 7, 1, 4, 11, 1, 2, 1, 1, 6, 1, 1, 1, 2, 1, 3, 1, 1, 4, 4, 12, 1, 7, 13, 3, 1, 6, 1, 16, 1, 24, 10, 1, 3, 3, 1, 6, 1, 3, 1, 99, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 108, 8, 2, 17, 1, 1, 1, 1, 28, 1, 3, 11, 1, 1, 11, 4, 1, 8, 1, 3, 1, 8, 1, 1, 1, 1, 2, 2, 2, 52, 5, 3, 1, 1, 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1, 2, 2, 3142, 4, 1, 1, 3, 4, 8, 2, 5, 1, 1, 21, 1, 4, 1, 2, 1, 23, 16, 1, 2, 24, 1, 2, 9, 1, 1, 3, 7, 3, 2, 1, 2, 6, 2, 5, 7, 1, 12, 5, 1, 11, 1, 3, 2, 9, 1, 1, 1, 1, 1, 2, 178, 1, 3, 6, 1, 1, 1, 2, 1, 17, 1, 3, 16, 3, 1, 48, 5, 1, 4, 1, 5, 1, 3, 5, 2, 6, 1, 7, 5, 1, 1, 1, 12, 1, 6, 1, 5, 2, 1, 1, 1, 3, 3, 1, 66, 1, 1, 13, 1, 7, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 4, 1, 9, 36, 2, 1, 4, 1, 1, 1, 1, 5, 2, 16, 6374, 12, 46, 4, 26, 2, 2, 2, 4, 1, 7, 2, 5, 1, 2, 30, 3, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 2, 1, 1, 5, 6, 3, 54, 1, 1, 17, 1, 6, 2, 2, 2, 7, 2, 2, 1, 2, 6, 5, 1, 3, 2, 3, 2, 1, 1, 1, 1, 1, 2, 6, 4, 7, 5, 5, 1, 2, 1, 2, 1, 3, 2, 8, 2, 2, 1, 9, 1, 55, 22, 1, 1, 14, 6, 1, 3, 1, 16, 5, 5, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 1, 1, 4, 2, 1, 3, 143, 1, 3, 2, 3, 1, 1, 12, 9, 22, 21, 2, 1, 22, 3, 80, 1, 3, 2, 1, 1, 2, 7, 28, 8, 4, 15, 1, 5, 3, 1, 2, 1, 13, 3, 2, 2, 32, 1, 1, 2, 34, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 11, 1, 1, 1, 18, 6, 1, 3, 2, 1, 3, 6, 3, 2, 2, 1, 1, 2, 4, 10, 2, 1, 2, 2, 1, 1, 2, 1, 3, 42, 1, 16, 1, 2, 2, 2, 2, 4, 1, 1, 2, 1, 14, 8, 1, 1, 1, 4, 1, 2, 2, 2, 7, 2, 5, 2, 1, 1, 6, 1, 1, 8, 1, 34, 1, 2, 2, 7, 17, 5, 1, 1, 1, 5, 1, 3, 2, 3, 1, 3, 4, 1, 13, 5, 1, 5, 2, 1, 3, 1, 2, 4, 12, 3, 1, 4, 4, 2, 1, 1, 15, 1, 1, 5, 1, 1, 2, 2, 1, 1, 9, 6, 5, 2, 1, 2, 12, 4, 1, 4, 1, 1, 4, 1, 1, 1, 1, 21, 12, 1, 13, 1, 14, 1, 2, 1, 1, 6, 2, 2, 3, 1, 2, 3, 1, 2, 2, 4, 1, 1, 1, 5, 2, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 33, 1, 1, 2, 5, 1, 5, 10, 1, 4, 1, 355, 1, 4, 1, 2, 19, 1, 6, 1, 9, 1, 13, 1, 4, 2, 1, 3, 2, 2, 8, 1, 35, 2, 4, 1, 1, 1, 1, 2, 3, 4, 1, 1, 9, 1, 1, 5, 1, 5, 2, 11, 3, 2, 1, 11, 4, 1, 1, 1, 1, 3, 3, 47, 2, 1, 4, 1, 1, 1, 1, 1, 10, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 113, 9, 1, 2, 1, 5, 4, 1, 1, 4, 2, 1, 2, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 5, 7, 4, 2, 1, 2, 1, 13, 188, 1, 3, 1, 1, 2, 5, 2, 4, 1, 2, 1, 1, 3, 1, 2, 5, 1, 3, 6, 3, 3, 1, 1, 20, 22, 1, 1, 9, 1, 1, 2, 1, 8, 92, 42, 2, 32, 1, 1, 16, 53, 3, 3, 2, 4, 17, 1, 18, 1, 3, 1, 178, 7, 3, 9, 1, 3, 3, 2, 5, 3, 1, 3, 1, 1, 2, 2, 1, 22, 23, 1, 4, 1, 39, 1, 1, 2, 1, 6, 1, 21, 1, 1, 4, 1, 1, 1, 10, 2, 1, 2, 4, 4, 6, 2, 1, 9, 85, 1, 1, 8, 37, 1, 3, 1, 1, 2, 1, 20, 1, 2, 1, 19, 4, 11, 5, 3, 2, 1, 1, 7, 1, 10, 2, 28, 2, 4, 2, 1, 4, 3, 1, 3, 7, 1, 2, 2, 2, 1, 1, 1, 2, 1, 18, 2, 1, 1, 1, 8, 2, 6, 10, 2, 4, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 32, 1, 3, 15, 1, 4, 5, 1, 2, 2, 3, 1, 7, 7, 3, 1, 3, 1, 1, 1, 16, 2, 1, 1, 3, 25, 1, 1, 18, 4, 3, 21, 7, 5, 7, 2, 393, 3, 3, 2, 6, 29, 1, 2, 18, 5, 1, 71, 1, 1, 2, 1, 7, 2, 2, 14, 10, 1, 3, 16, 1, 6, 7, 2, 2, 3, 12, 2, 8, 25, 1, 1, 1, 2, 21, 1, 1, 3, 4, 6, 8, 119, 1, 3, 6, 1, 69, 2, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 6, 1, 5, 3, 11, 2, 2, 1, 1, 87, 1, 3, 2, 3, 10, 1, 1, 1, 3, 1, 7, 8, 1, 6, 1, 2, 2, 1, 1, 1, 2, 9, 1, 4, 170, 3, 1, 2, 1, 355, 3, 13, 4, 1, 1, 2, 3, 1, 5, 7, 5, 11, 2, 1, 10, 1, 76, 2, 1, 2, 1, 4, 1, 1, 1, 1, 2, 4, 5, 7, 21, 3, 1, 2, 2, 70, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 7, 20, 1, 196, 1, 3, 3, 1, 10, 4, 1, 5, 1, 3, 10, 2, 14, 3, 2, 1, 1, 27, 1, 1, 1, 143, 1, 1, 10, 2, 21, 1, 1, 2, 4, 2, 27, 1, 2, 14, 1, 1, 2, 1, 1, 6, 3, 1, 1, 45, 1, 2, 7, 4, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 3, 19, 3, 86, 1, 1, 1, 217, 7, 1, 1, 1, 1, 2, 2, 16, 1, 10, 1, 2, 1, 2, 1, 8, 2, 7, 9, 1, 2, 5, 7, 1, 7, 14, 5, 1, 5, 2, 2, 3, 3, 1, 1, 3, 3, 3, 1, 1, 13, 1, 11, 8, 1, 8, 2, 2, 2, 2, 1, 4, 1, 1, 1, 1, 4, 5, 4, 1, 3, 1, 1, 1, 1, 12, 18, 4, 2, 1, 1, 141, 4, 14, 3, 6, 6, 4, 4, 2, 1, 3, 9, 1, 12, 3, 2, 5, 4, 6, 1, 16, 5, 3, 1, 1, 1, 17, 4, 1, 1, 1, 31, 9, 1, 2, 1, 1, 167, 1, 1, 2, 1, 1, 2, 2, 1, 4, 9, 1, 31, 6, 198, 2, 24, 2, 1, 6, 1, 18, 1, 2, 2, 1, 7, 1, 1, 1, 3, 31, 1, 2, 2, 52, 2, 5, 3, 1, 1, 2, 2, 1, 7, 7, 1, 6, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 5, 2, 1, 12, 1, 4, 39, 1, 4, 2, 6, 17, 1, 2, 4, 1, 11, 2, 2, 4, 1, 5, 2, 1, 13, 2, 21, 4, 1, 23, 2, 3, 27, 7, 3, 48, 8, 1, 1, 18, 12, 1, 25, 1, 1, 1, 16, 1, 4, 3, 1, 6, 2, 3, 2, 5, 2, 1, 6, 12, 3, 2, 6, 1, 3, 4, 1, 5, 1, 2, 1, 9, 2, 36, 2, 2, 1, 4, 2, 12, 16, 1, 1, 29, 1, 3, 4, 3, 1, 1, 2, 1, 22, 1, 2, 1, 1, 2, 1, 2, 13, 1, 3, 2, 23, 13, 5, 1, 4, 1, 5, 1, 358, 1, 11, 1, 1, 1, 2, 1, 5, 5, 1, 15, 2, 16, 2, 21, 1, 2, 2, 2, 5, 1, 17, 4, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 112, 1, 17, 1, 8, 3, 1, 1, 1, 26, 1, 1, 10, 12, 59, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 130, 1, 1, 1, 1, 1, 17, 1, 1, 3, 17, 1, 2, 2, 9, 1, 4, 1, 18, 1, 1, 7, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 5, 3, 50, 1, 4, 1, 13, 21, 1, 1, 1, 3, 1, 1, 1, 7, 5, 3, 1, 6, 1, 2, 1, 1, 1, 3, 5, 1, 2, 2, 3, 1, 7, 13, 2, 3, 1, 3, 27, 2, 3, 3, 1, 27, 2, 1, 15, 1, 12, 185, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 123, 8, 1, 2, 170, 1, 1, 6, 1, 8, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 3, 13, 1, 15, 36, 1, 7, 3, 5, 8, 1, 3, 3, 1, 1, 4, 5, 11, 1, 21, 3, 2, 8, 2, 1, 4, 1, 5, 14, 2, 4, 1, 1, 1, 1, 10, 1, 1, 12, 4, 1, 3, 1, 2, 3, 1, 5, 1, 1, 5, 1, 1, 1, 2, 74, 3, 1, 9, 2, 1, 31, 3, 1, 1, 4, 1, 1, 2, 4, 1, 1, 1, 17, 4, 1, 2, 1, 1, 1, 1, 1, 48, 6, 2, 2, 1, 2, 4, 8, 1, 21, 3, 1, 7, 1, 3, 2, 1, 1, 4, 2, 2, 1, 1, 4, 1, 2, 11, 4, 1, 3, 4, 4, 1, 1, 14, 2, 5, 2, 2, 5, 2, 2, 1, 10, 2, 1, 3, 1, 1, 1, 6, 3, 11, 3, 2, 3, 1, 1, 2, 142, 1, 4, 1, 7, 56, 7, 1, 2, 5, 1, 8, 1, 1, 3, 1, 1, 3, 20, 5, 4, 3, 14, 2, 15, 1, 2, 1, 3, 2, 1, 1, 3, 11, 4, 1, 1, 1, 4, 1, 13, 10, 3, 2, 13, 1, 1, 365, 1, 2, 1, 1, 1, 4, 32, 9, 2, 15, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 6, 3, 8, 2, 5, 3, 1, 1, 1, 9, 1, 2, 1, 6, 1, 2, 1, 1, 7, 1, 3, 2, 1, 1, 14, 1, 2, 1, 3, 1, 1, 2, 14, 1, 8, 1, 21, 1, 121, 1, 1, 2, 1, 1, 1, 2, 7, 2, 1, 1, 4, 1, 1, 1, 1, 3, 3, 1, 3, 2, 1, 6, 1, 1, 2, 2, 19, 2, 2, 2, 3, 2, 1, 23, 2, 2, 1, 7, 1, 1, 4, 22, 2, 13, 1, 1, 2, 8, 1, 24, 1, 9, 4, 5, 1, 8, 7, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 55, 2, 1, 1, 1, 3, 1, 3, 4, 2, 4, 1, 5, 12, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 4, 1, 10, 1, 1, 1, 1, 3, 10, 5, 2, 1, 6, 4, 1, 1, 3, 4, 1, 2, 16, 2, 1, 1, 5, 1, 6, 1, 1, 1, 10, 1, 4, 1, 3, 5, 1, 1, 1, 4, 6, 1, 1, 3, 9, 1, 7, 1, 1, 1, 4, 76, 1, 12, 18, 2, 1, 1, 9, 2, 7, 18, 1, 1, 1, 1, 9, 2, 1, 1, 4, 3, 1, 6, 3151, 1, 6, 13, 2, 1, 1, 3, 1, 25, 1, 2, 1, 2, 4, 1, 3, 1, 2, 1, 10, 3, 1, 2, 1, 13, 1, 3, 11, 2, 1, 1, 5, 4, 1, 1, 2, 1, 2, 44, 3, 9, 2, 3, 1, 1, 7, 2, 2, 3, 11, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 6, 2, 4, 6, 2, 1, 4, 4, 1, 2, 1, 2, 1, 4, 164, 3, 1, 2, 4, 15, 8, 1, 3, 1, 2, 1, 3, 139, 1, 4, 75, 7, 2, 1, 1, 1, 14, 2, 234, 2917, 2, 1, 1, 1, 23, 1, 2, 2, ...]

RM with a view: An exercise in self-referential analysis.

While Khinchin's constant has been defined as the limit of a geometric mean, that constant's simple continued fraction should not be exempt from the principle it embodies. The (above) number (cfKhinchin) is unique in the sense that it is the only number governed, if you will, by the number it represents. I will use the shorthand "RM" to mean "the Reduced geometric Mean of the first n terms (excluding the zeroth term '2') of cfKhinchin", i.e., the mean minus Khinchin's constant.

RM 1-300:

Zoom (140-180):

The value at 148 is 0.000453439. No value closer to zero will appear until 5920.

Zoom (250-290):

The value at 275 is -0.000598811.

RM 300-6000:

Zoom (5915-5965):

The value at 5920 is 0.000087387. The value at 5943 is 0.000016205.

RM 6000-100000:

One More Zoom (8370-8410):

The value at 8381 is 2.27347 *10^-6. The value at 8401 is -3.46004 *10^-7. No value closer to zero will appear until 89953.

Summary:

Overall, the points that yield progressively better approximations to Khinchin's Constant [i.e., the number itself] are:

{1, -1.68545}, {2, -1.27124}, {3, -0.531017}, {15, -0.451007}, {23, -0.355948}, {26, -0.297795}, {81, -0.293319}, {104, -0.221656}, {109, -0.216905}, {111, -0.198337}, {120, -0.174483}, {127, -0.163783}, {135, -0.11783}, {136, -0.0988251}, {141, -0.0863652}, {142, -0.0837382}, {144, -0.0766779}, {145, -0.0416949}, {146, -0.0215688}, {147, 0.0185668}, {148, 0.000453439}, {5920, 0.000087387}, {5943, 0.000016205}, {8381, 2.27347 *10^-6}, {8401, -3.46004 *10^-7}, {89953, -2.17726 *10^-7}, {91368, 5.46572 *10^-8}

...the first value of each pair representing the sequence A048613.

12 January 2000 © Rarebit Dreams