[vbffnrxd] continued fraction expansion of truncated sum of prime reciprocals

length after 10000 primes, Mathematica:

Length[ContinuedFraction[Total[1/Prime[Range[10000]]]]]

88444

(note: the sum diverges, very slowly, when taken over all primes.)

number of continued fraction terms grows faster then linearly.  conjecture n*log(n).

FromContinuedFraction can check that the sum matches.

until termination, the terms don't look unusual, so this can create a rational number that initially looks like a typical irrational number (Khinchin's constant).  (but sampling a finite number of random digits past the decimal point probably also works just as well.)

ContinuedFraction[Total[1/Prime[Range[100]]]]

{2, 9, 2, 2, 10, 1, 2, 12, 14, 1, 8, 1, 1, 18, 1, 1, 5, 1, 1, 5, 5, 1, 4, 1, 10, 1, 3, 3, 2, 24, 2, 1, 3, 1, 5, 6, 5, 1, 58, 1, 1, 2, 1, 2, 6, 3, 1, 2, 3, 1, 1, 2, 3, 1, 34, 1, 2, 1, 2, 2, 1, 19, 1, 7, 1, 16, 7, 14, 1, 1, 1, 18, 1, 2, 6, 1, 1, 1, 2, 6, 6, 3, 1, 4, 34, 408, 1, 1, 1, 87, 85, 1, 45, 1, 1, 3, 12, 1, 2, 3, 2, 1, 4, 6, 27, 1, 1, 6, 1, 1, 539, 2, 2, 1, 2, 1, 1, 1, 3, 3, 192, 10, 167, 2, 6, 7, 2, 1, 4, 1, 1, 1, 1, 1, 3, 13, 2, 5, 1, 10, 2, 2, 3, 1, 119, 14, 2, 1, 3, 1, 2, 1, 1, 3, 11, 13, 2, 4, 3, 13, 165, 1, 4, 1, 1, 1, 3, 1, 7, 1, 5, 3, 15, 1, 1, 181, 1, 1, 3, 1, 1, 9, 1, 1, 4, 3, 1, 1, 4, 1, 2, 6, 2, 2, 11, 1, 4, 5, 15, 3, 1, 6, 2, 9, 1, 1, 1, 5, 14, 3, 2, 1, 1, 1, 1, 2, 1, 2, 126, 1, 2, 1, 1, 1, 3, 1, 82, 8, 1, 1, 11, 24, 4, 1, 2, 1, 3, 19, 2, 1, 1, 1, 1, 1, 29, 1, 2, 3, 1, 1, 9, 2, 7, 1, 3, 1, 4, 1, 1, 2, 24, 1, 6, 6, 2, 4, 9, 1, 3, 1, 1, 1, 42, 1, 20, 2, 2, 1, 1, 1, 2, 6, 1, 4, 1, 22, 42, 1, 142, 2, 4, 2, 1, 5, 39, 1, 5, 1, 5, 1, 9, 1, 5, 1, 1, 4, 1, 13, 26, 1, 1, 51, 1, 58, 1, 1, 4, 9, 14, 1, 7, 20, 2, 1, 2, 1, 1, 1, 3, 1, 6, 9, 1, 1, 1, 67, 4, 2, 1, 1, 3, 3, 1, 4, 2, 3, 2, 1, 1, 6, 3, 1, 10, 2, 1, 8, 1, 1, 1, 1, 99, 70, 2, 1, 1, 1, 4, 1, 10, 9, 5, 1, 1, 1, 1, 1, 3, 36, 1, 19, 69, 5, 110, 1, 9, 1, 1, 2, 2, 1, 15, 2, 5, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 3, 3}

407 terms.