A064442 Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...

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, 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, 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

Offset: 1

Author

Robert G. Wilson v, Oct 01 2001

Keywords

• cons
• nonn

Comments

Continued fraction expansion of the prime numbers. - Harvey P. Dale, Sep 25 2012

Examples

			2.313036736433582906383951602641782476396689771803256340210124442144564731776...
		

Links

• MathOverflow, What is the effect of adding 1/2 to a continued fraction?

Crossrefs

Cf. A060997, A302937.

Programs

Mathematica

RealDigits[ N[ FromContinuedFraction[ Table[ Prime[n], {n, 1, 100} ]], 100]] [[1]]
RealDigits[FromContinuedFraction[Prime[Range[200]]],10,120][[1]] (* Harvey P. Dale, Sep 06 2021 *)

Formula

1/A084255. - Franklin T. Adams-Watters, Jul 31 2009

From Peter Bala, Nov 26 2019: (Start)

Denoting the constant by c we have the related simple continued fraction expansions (prime(n) denotes the n-th prime number):

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, ...];

(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, ...];

(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)