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%I #26 Sep 06 2021 15:03:26
%S 2,3,1,3,0,3,6,7,3,6,4,3,3,5,8,2,9,0,6,3,8,3,9,5,1,6,0,2,6,4,1,7,8,2,
%T 4,7,6,3,9,6,6,8,9,7,7,1,8,0,3,2,5,6,3,4,0,2,1,0,1,2,4,4,4,2,1,4,4,5,
%U 6,4,7,3,1,7,7,6,2,7,2,2,4,3,6,9,5,3,2,2,0,1,7,2,3,8,3,2,8,1,7,4,5,3,0,1,5,8,2
%N Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...
%C Continued fraction expansion of the prime numbers. - _Harvey P. Dale_, Sep 25 2012
%H MathOverflow, <a href="https://mathoverflow.net/questions/128676/what-is-the-effect-of-adding-1-2-to-a-continued-fraction">What is the effect of adding 1/2 to a continued fraction?</a>
%F 1/A084255. - _Franklin T. Adams-Watters_, Jul 31 2009
%F From _Peter Bala_, Nov 26 2019: (Start)
%F Denoting the constant by c we have the related simple continued fraction expansions (prime(n) denotes the n-th prime number):
%F 2*c = [4; 1, 1, 1, 2, 14, 5, 1, 1, 6, 34, 9, 1, 1, 11, 58, 15, 1, 1, 18, 82, 21, ..., 1, 1, (prime(3*n) - 1)/2, 2*prime(3*n+1), (prime(3*n+2) - 1)/2, ...];
%F (1/2)*c = [1; 6, 2, 1, 1, 3, 22, 6, 1, 1, 8, 38, 11, 1, 1, 14, 62, 18, 1, 1, 20, 86, 23, ..., 1, 1, (prime(3*n+1) - 1)/2, 2*prime(3*n+2), (prime(3*n+3) - 1)/2, ...];
%F (c + 1)/(c - 1) = [2; 1, 1, 10, 3, 1, 1, 5, 26, 8, 1, 1, 9, 46, 14, 1, 1, 15, 74, 20, ..., 1, 1, (prime(3*n+2) - 1)/2, 2*prime(3*n+3), (prime(3*n+4) - 1)/2, ...]. (End)
%e 2.313036736433582906383951602641782476396689771803256340210124442144564731776...
%t RealDigits[ N[ FromContinuedFraction[ Table[ Prime[n], {n, 1, 100} ]], 100]] [[1]]
%t RealDigits[FromContinuedFraction[Prime[Range[200]]],10,120][[1]] (* _Harvey P. Dale_, Sep 06 2021 *)
%Y Cf. A060997, A302937.
%K cons,nonn
%O 1,1
%A _Robert G. Wilson v_, Oct 01 2001
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